Curso 3 de preparación en GMAT

1. Fractions Introduction: a fraction consists of two parts. A numerator, the top part and the denominator. The bottom part. If the numerator is smaller than the denominator, the fractions called a proper fraction, and its value is less than one. For example, 1/2 for fists and three over pie are all proper fractions and therefore less than one. If the numerator is larger than the Dominator, then the fraction is called improper and its value is greater than one. For example, three House five Force or PI over three are all improper fractions, and there for greater than one, an improper fraction can be converted into a mixed fraction by dividing its denominator into this numerator, for example, since two divides into 73 times with remainder of one we get seven halves equals three and 1/2. Now we can show the long division dividing two into seven. It goes in there three times, subtracting six or seven. You get one, so we get three and 1/2 to convert a mixed fraction into an improper fraction. Multiply the denominator and the fraction so you multiply the five in the three, as we do here, and then add the numerator add the plus two, then write all of it over three in a negative fraction. The negative symbol can be written on the top, in the middle or on the bottom. However, when a negative symbol appears on the bottom, it is usually moved to the top or the middle of the fraction. If both terms in the denominator of a fracture negative, negative, simple is often factored out and moved to the top or the middle of the fraction. So here, both terms and negative X and the negative to our negative. So you factor out and negative and then write it either in front of fraction or in the top of the fraction. 2. Fractions Cross multiplying: to compare two fractions. Cross multiply. The larger number will be on the same side as a larger fraction. So if we have a over B versus C over D, we cross multiply. And that gives us a D on the left versus BC on the right. Now, if the product BC is greater than the product a D than the fractions C over D is larger than the fraction a over B. 3. Fractions Example 1: cross multiplying. We get nine times 11 on the left side and 10 times 10 on the right side. Nine times 11 is 99 and 10 times 10 is 100. Now 100 is greater than 99. Hence, 10 11 is greater than nine tense. 4. Fractions Reducing: always reduce a fraction to its lowest terms. For example, if we had X squared plus X over X plus one, then when you factor out an X on the top and then cancel the common factor X plus one. 5. Fractions Example 2: factory, not the common factor of two in the numerator we get now, the factors of one or one and one and one plus one is two, which is the middle term. So it factors into X plus one and X plus one over the X plus one square canceling the X plus ones here in here we get to hence the answer is C. 6. Fractions Solving by Multiplying by the LCD: to solve a fractional equation, multiply both sides by the l c D. The lowest common denominator to clear fractions. 7. Fractions Example 3: the question. What is the value of X in terms of? Why means that we need to solve the equation for acts in terms of why no words we need the right X equals G of why and noticed exes on both the top and the bottom of the fractions, so multiplied by X minus three to clear the fraction and canceling X minus threes gives us X Plus three is equal toe y times X minus three. Distribute the why on each term we get X Y minus three wide on the right and X plus three on the left. Now we're solving for X, so we want to get all the exits on one side. So subtract X y from both sides of the equation and reminds most of practice three from both sides to move it to the other side. And that gives us explain the sex y equals minus three Y minus three. Canceling the x y. You know, we have a common factor of X. Factoring it out. We get one minus y and then finally dividing both sides by one minus why we get X equals and just notice that this is answer choice D 8. Fractions Complex Fractions: when dividing a fraction by a whole number or vice versa. You must keep track of the main division bar for this first problem. The main division bars on the top. So by the definition of division, we write down the numerator A and we invert or reciprocate the denominator that is a C over B or a C over B. Now, for the second example, the main division bar is on the bottom. So we write down the new meter a over B as is, and we flipped the sea over and often will write C over one for symmetry. With the top of the fraction. So see, over one becomes one oversee, then we get a over B. C. 9. Fractions Example 4: Let's first get a common denominator in the top of the fraction. Multiplying top and bottom of one by two, we get tu minus one over to all divided by three. Where this is our main division bar and simplifying the top. We get 1/2 over three and for symmetry, weaken right 3/1. Now dividers to invert. Multiply so right down the numerator, which is 1/2. Then we take the denominator, which is 3/1 and reciprocated. Flip it over and multiply, which gives us 16 hence, the answer is D. 10. Fractions Example 5: getting a Tom common denominator in the bottom of the fraction we multiply top and bottom of why by Z, which gives us one main division bar over Y Z minus one over the common denominator Z. And this is our cup or may. This is our main division bar here. So to divide means to invert multiplies, we get one times denominator flipped over and then dropping the one we get Z over Y Z minus one. Now this notice that that is answer choice d. 11. Fractions Multiplying: multiple infractions. His routine merely multiply the numerator, the top part of the fractions and multiply the denominators the bottom part of the fraction . So a over B times C over D. We just simply multiply the and see and the BND, for example, 1/2 times three force is one times three and then two times for which gives three AIDS. 12. Fractions Adding by Cross multiplying: two fractions can be added quickly by cross multiplying. So here we multiply A and D and then B and C over the product of B and E. 13. Fractions Example 6: cross multi plane. We get one times four minus two times, three over two times four. Simplifying gives us four minus six Hillary or negative to over eight, which reduces to negative 1/4. Hence the answer is C. 14. Fractions Example 7: recall that the average is the sum of the expressions divided by the number of expressions . Since we have two expressions X and one over acts, we get X Plus one over X, divided by two. Getting in a common denominator in the top multi top bottom of X by axe, which gives his X squared plus one over X Main division bar over to and for symmetry, weaken right to over one and to divide beings to invert, multiply so copy down the numerator and take the denominator and flip it over. Which gives us what times 1/2. And that could be written a little more compactly as X plus one over two X and now just told us that that is answer choice be. 15. Fractions Adding by Getting a Common Denominator: to add three or more fractions with different denominators, you need to form a common denominator of all the fractions. For example, to add the fractions in this expression, we have to change the denominator of each fraction into the common denominator. 36 No. 36 is a common denominator because 34 and 18 all divided into it evenly. This is done by multiplying the top and bottom of each fraction by an appropriate number. This does not change the value of expression because any number divided by itself is one. So for the 1/3 remote by top and bottom by 12 to turn it into a denominator of 36 for the four problem morning by nine to get 36 an 18 top and bottom I to to get 36 and in these numbers up we get 23 36. You may remember from algebra that to find a common denominator of a set of fractions, you prime factor the denominators and then select each factor the greatest number of times it occurs in any of the factories ations. However, that is too cumbersome. A better way is to simply add the largest denominator to itself until the other denominators divide into it evenly. In the above example, we just add 18 to itself and get 36 then note that both three and four go into 36 evenly. Therefore, 36 is the least common denominator. 16. Fractions Finding a Common Denominator: to find a common denominator of a set of fractions. Simply add the largest, a nominator to itself, until all the other denominators divide into it evenly. For example, if we had 1/2 1/3 and 1/8 we want to get a common denominator between 23 and eight. We take the largest number eight and add it to itself, which gives us 16 now to goes into 16 evenly. But three does not. So he add 8 to 16 again, and we get 24 and two goes in 24 as this three and 24 is the least common denominator. 17. Fractions Unusual Behavior: fractures often behave in unusual ways. For example, when square in a number we usually expected to get larger. But with the proper fraction, it will actually become smaller and taking the square root of a number, we normally expect to make it smaller. With the proper fraction, the square root will become larger. And this is true only for proper fractions. That is, fractions between zero and one. For example, 1/3 squared is 1/9 and 1/9 is smaller than 1/3 and the square root of 1/4 is 1/2 and 1/2 is bigger than 1/4. 18. Decimals Introduction: If a fractions denominator is a power of 10 it can be written a special form called a decimal fraction. Some common decimals are 1/10 which is 0.1 to one hundreds, which is 10.2 and 31 thousands, which is 310.3 Notice that the number of decimal places corresponds to the number of zeros in the denominator of the fraction. Here we have three decimal places for the corresponding three zeros in the denominator. Also note that the value of the decimal place decreases to the right of the decimal point. So 4.1234 we get tense hundreds, thousands and then 10 thousands. This decimal can be written in expanded form as follows. Sometimes a zero is placed before the decimal point to prevent misreading the decimal as a whole. Number zero has no effect and no mathematical meaning on the value of the decimal. For example, point to is written as 0.2 fractions can be converted into decimals by dividing the denominator, the bottom into the numerator, the top of the fraction. For example, to convert 5/8 to a decimal divide eight into five, as shown below 19. Decimals Adding, Subtracting, Multiplying, Dividing: the procedures for adding subtracting, multiplying and dividing decimals are the same as for whole numbers, except for a few small adjustments adding and subtracting decimal to add or subtract, decimals merely aligned the decimal points and then add or subtract as you would with whole numbers, as shown in these examples here, multiplying decimals multiplied decimals as you would with whole numbers. The answer will have as many decimal places as the some of the number of decimal places in the numbers being multiplied. For this example, we have two decimal places in the top decimal and one in the bottom, so we get a total of three decimal places. So for the answer, recount over 1 to 3 decimal places dividing decibel before dividing decimals, moved the decimal point of the divisor all the way to the right and move the decimal point of the dividend, the same number of spaces to the right, adding zeros if necessary, then divide as you would with whole numbers. So here we have two decimal places in the 20.24 so we move it over to places and we get 24 and the 0.6 is moved over two decimal places and we have to add a zero when we do so and then we add some zeros afterwards and you can have as many Asian need in this case we needed only one. 20. Decimals Example 1: recall that percent means to divide by 100 so 0.1% equals 0.1 over 100 which is 0.1 now to convert 1/5 into a decimal, we divide five into one. It was at a zero. Five doesn't go into zero, but five does go into 10 twice, so we get a two. And that's after the decimal point, because the zeros after the decimal point we get 10. Subtract and you get a remainder of zero No, when dealing with percent of beings multiplication. So we have point to times point zeroes 01 and perform that multiplication vertically. We get 0.2 times 0.1 which gives us 0.0 zero to hence the answers E. And you might be surprised that the G R E would probably consider this to be a hard problem , even though it involves only elementary arithmetic operations. Never last. Most students would probably miss it 21. Decimals Example 2: this first convert this decibel into a fraction. It is one over 1000 and when over 1000 can be written as 1/10 cubed which in turn could be written is 10 to the negative three and this is cubed. So we get 10 to the negative three cubed, which is 10 to the negative ninth power. Now we formed the ratio between the larger number 0.1 and the smaller number 10 to the negative. Ninth Power 0.1 is 1/10 over 10 to the negative ninth power and on the top we get 10 to the negative one and then subtracting the bottom exponent from the top exponents, we get 10 to the negative one, minus negative nine or 10 to the negative one plus nine, which is 10 to the His 0.1 is larger than 0.1 to the third power by a factor of 10 to the eighth. And the answer is D 22. Decimals Example 3: Let's comfort 0.99 into a fraction 99 over 100. Now this is a proper fraction greater than zero unless than one. Hence, if we take the square root of it, it will get bigger. And if we square it, you'll get smaller. And this is X. This is why. And this is easy, which is choice see. 23. Fractions and Decimals Problem 1: keeping track of the main division bar. We invert that it on mentor and multiply it by the numerator. So we get two times 3/4, which is 6/4 and cancelling the common factor of two. We get 3/2. Hence the answer is C. 24. Fractions and Decimals Problem 2: Let's start with A and B. It's 56 versus 4/5. Cross small plane. We get 25 versus 24 says 25 is greater than 24. A is larger than be turning to see. We get 56 versus 1/2 cross Small plane gives us 10 versus 6 10 is greater than six. So again, choice say is larger and continuing. This matter will see that Choice A is also larger than D N. E. Hence the answer is a. 25. Fractions and Decimals Problem 3: Let's factor all the expressions and then cancel it. We can. This expression here is a perfect square. Try no meal so it could be factored into X plus three times X plus three over the original X Plus three. And here we have a difference of squares because nine is three square and it factors into X plus three an X minus three over the original X minus three. Canceling we're left with X plus three times X plus three, which is X plus three square. Hence the answer is C. 26. Fractions and Decimals Problem 4: getting a common denominator. We replace one with 3/3, which gives us 1/4 minus three over three in the main division. Bars on the top, which reduces the 1/4 minus three, is one over three, and again the main division bar is on. The tops will be right down the top and reciprocate the bottom of the fraction, which gives us 3/1, which is three hence, the answer is D. 27. Fractions and Decimals Problem 5: since squared a fraction that is between 01 makes it smaller. We know that statement one is true and this eliminates B and C. Likewise, take the square root of a fraction that is between zero and one makes it larger, not smaller. So statement three is false and that eliminates E now for statement to use u substitution. We need check only one number in this retrain range because all we have is proper fractions between zero on one choosing XP 1/2 we get notice. The main division bars is the top are here. So we inverted, multiply and you get one times for over one, which is four. And we know that 1/2 going back to the expression we have 1/2 which is the X that we chose is, in fact less than four. And that is a true statement. Therefore, the answer is D 28. Fractions and Decimals Problem 6: Let's just start reciprocating the numbers. The reciprocal of one is 1/1, which is one so statement. One is true. One is the reciprocal of itself. For statement to let's start with the second number, it'll be easier. The reciprocal of negative 11 is one over negative 11 which is negative 1/11 which is not equal to 1/11 because this is positive and that's negative. So statement, too, is false. The reciprocal of one of radical five is one over radical. Five. It doesn't look like the other number, but let's rationalize it. And that gives us radical five on the top and radical five times. Article five is five on the bottom, which is the other number. Hence statement three is true. Therefore, the answer is D 29. Fractions and Decimals Problem 7: Let's start at the bottom of the complex, fresh and getting a common denominator. We replace one with 2/2, which gives us tu minus one is one over to, and 1/1 half is one times the 1/2 reciprocated. Which, of course, is to over one simplifying. We get 1/1 minus two, which is one over negative one, which is negative one hence the answer is B. 30. Fractions and Decimals Problem 8: notice that we have nine factors of 10 here and 10 factors of 10 here. So we have an extra factor of 10 in this term. So let's peel off one of the tens peeling off one of the tens. We get this expression because one plus nine gives us back the 10. Now, just notice that there is a common factor here of 1/10 to the ninth Power. Factoring that out, we good now something has to remain in the first term, and that's a one minus one over the 10. Now replace one with 10/10 to get a common denominator and 10 minus one is nine. So we have 9/10 and then we'll play in the tens. We get 9/10 to the 10th which is Choice D. 31. Fractions and Decimals Problem 9: first writeth e expression in the denominator over one. To balance out the complex fraction, copping down the top and invert and inverting and multiplying by the bottom. We get one over two times X plus one him factor two out of the top expression here. Now here we have a difference of squares so we can factor it into X plus one an X minus one over the original X minus one. Now cancel the X minus ones and the X plus ones, and we get to over two, which is one hence the answers be. 32. Fractions and Decimals Problem 10: First we calculate the value of kill notice. One is in peas position. So everywhere that P appears in the form that we replace it with one. So we get one star is equal to 1/2 Main division bar four times one minus one, which is 1/2 over three and again for cemetery weaken. Right, This is 3/1. Inverting the multiplying. We get 1/2 times 1/3 which is 16 Now that we know the value of Q, we plug 1/6 into the formula that is, replace all appearances of P with 16 So 16 star is equal to 16 divided by two Main division bar four times 16 minus one on the top, we get 16 times, 1/2 and this is for six on the bottom play, this one and 46 reduces to 2/3. So on the top we get 1/12 and the bottom. We have 2/3 minus one, which is negative 1/3 inverting the multiplying. We get 1/12 times negative 3/1, which reduces to negative 1/4. And now just noticed that that is answer choice. See 33. Fractions and Decimals Problem 11: Let's start by getting a common denominator in the bottom of the fraction will apply in top and bottom of why by Z We get now the main division bars on the top. So we invert multiplied and now we multiply Top model bite of acts by Z y minus one to get a common denominator and distributed excuses xz Why minus x on the top, minus c all over the common denominator. And now we just noticed that this is answer choice d. 34. Fractions and Decimals Problem 12: forming the negative reciprocal of the expression gives us negative one over. Now let's get a common dominator on the bottom. So multiply top bottom of X by y and moved by top and bottom of one over. Why, by axe That gives us why. Plus acts over. That's why now remain division bars on the top. So we invert the multi the bottom and multiply. So you get negative one times x y over y plus X or to simplify. We get negative. X y over y plus x. I don't know now noticed that this is answer choice E. 35. Equations Introduction 1: when simplifying algebraic expressions, we performed the operations within parentheses first, then exponents, then multiplication and then division and then addition. And lastly, subtraction. This can be remembered by the demonic pen does from left to right. Please excuse my dear Aunt Sally. When solving equations, however, we apply the demonic and reverse order sad map as we go from right to left. This is often expressed as follows inverse operations in inverse order. The golden solvent equation is to isolate the variable on one side of the equal sign, usually the left side. This is done by identifying the main operation, addition, multiplication, etcetera and then performing the opposite operation. 36. Equations Introduction 2: Let's solve the fallen equation for X. The main operation is addition. Remember, Addition now comes before multiplication. We're doing sad now, So subtracting wife from both sides yields track. Why here in here And the wise here add up to zero So they disappear. Now the only operation remaining is the multiplication between two and X. So we divide both sides of the equation by two. Canceling the twos gives us this expression. 37. Equations Introduction 3: that's all. The falling equation for X here X appears on both sides of equal sign. So let's move the X on the right side to the left side. But no sex is trapped inside. Parentheses here to release it will distribute the two over the parentheses, and that will give us two X minus 10. Now subtract two x from both sides of the equation, which gives us X minus four equals negative 10 and then finally adding four to both sides, We get X equals negative six. 38. Equations Properties of Equations 1: we often manipulate equations without thinking about what the equations actually say. The test writers like to test this oversight Equations are packed with information. Take, for example, the simple equation three X Plus two equals five. Since five is positive, we know that the Expression three X plus two must be positive as well. An equation means that the terms on either side of the equal sign are equal in every mathematical way. Hence any property. One side of an equation has the other side will have a swell following are some immediate deductions that can be made from simple equations. Here's since the difference between why and X is one which is a positive number. We know that why is greater than acts. And if y squared equals x squared that we know that why equals plus or minus X or the absolute value of Y equals absolute value. Vax, that is, except why can differ only in sign and by the way, that will be true for all, even exponents. And if y cube equals execute, then we can conclude that why equals X And again, that's true for all odd exponents. 39. Equations Properties of Equations 2: here we are told that Y is equal X squared. And we know all squares are greater than or equal to zero. Hence why is greater than or equal to zero? Here we have the quotient between why and X squared equals a positive number. One there for the UAE has speak positive. Otherwise the expression would be negative. Here we have the quotient between two odd exponents. Why, to the one versus X cube is positive. And since odd exponents preserve negative numbers, we know that both X and y must be positive or both. X and Y must be negative against, since squares are greater than or equal to zero. The only way the sum of two squares could equal zero if each of them is equal to zero. Otherwise, the expression would be greater than zero for this example, since we're told that X is positive, we know that why must be positive as well, because three times Y is equal to four times acts. And we also know that why is greater than X because we have to multiply X by a bigger number, then why, in order to make them equal each other now for the same expression. If X is less than zero, then why will be negative? M. I will be less than acts for this expression, since a radical is always greater than or equal to zero. We know that why is greater than equal to zero. And we also know that X is greater than negative, too, because you cannot take the square root of a negative number. So X plus two must be greater than or equal to zero. So backs must be greater than or equal to negative two. And if y is equal to two acts, then we know why is, even by definition and one more than an even number is an odd number. And if you have a practice to never setting with zero, we know that one of the must be zero or the other or both. 40. Equations Solving for Expressions: ineligible. You solve an equation for say why? By isolating why On one side of the quality symbol on the test, however, you are often asked to solve for an entire term for example, three minus y by isolating it on one side. 41. Equations Example 1: first, let's translate the claws into an equation. We have a plus three. A is translates as equals for less than something, namely the B plus three B notice. The minus four goes on this term. Because the B Plus three B is the larger term, many students mistakenly subtracted from the left side, combining like terms we get for a is equal to four B minus four. That's a practice for be from both sides for a minus four B equals negative for and then divide each turn by four. This gives is a minus B, which is the term we're looking for, and it will equal negative one. Hence the answer is B. 42. Equations 'Triple' Equations: Sometimes on the test, a system of three equations will be written as one long triple equation. For example, the three equations X equals why Weichel, Z and X equals E can be written more compactly as X equals Y equals E. 43. Equations Example 2: from this triple equation, we get three separate equations. W equals two acts. Two X is equal Teoh Radical to why and w is equal to radical to why? From the middle equation, we get X equals radical too over two. Why so x minus So w minus X equals radical to why minus X radical to over two. Why? Getting a common denominator? We multiply top bottom by two. So we get too radical too. Why? And these air light terms We have two of them here and one here. So we get a total of one one radical to why? Over to And now we just noticed that is answer choice be 44. Equations Adding Equations: often on the test, you can solve a system of two equations in two unknowns by merely adding or subtracting the equations instead of solving for one of the variables and then substituting it into the other equation. 45. Equations Example3: all the system of two equations by merely subtracting the equations this aligning vertically and subtract the P Square's cancel and we get minus two. Q Square is equal to negative eight. Divided by two you get to get Q squared is equal to four. Take the square root of both sides. We get Q equals plus and minus square root of four, which is plus and minus two. No negative two is not offered, but pause of two is. It is answer. Choice a. 46. Equations Method of Substitution: method of substitution the four step method, although in the test you can usually solve a system of two equations into unknowns by merely adding or subtracting the equations, you still need to know a standard method for solving these types of systems. The four step method will be illustrated with the following system. So one of the equations for one of the variables solving the top equation for why we get 10 minus two acts by simply subtracting two x from both sides now substitute this result into the other equation. So substituting it into the bottom equation, we get five x minus two times the quantity 10 minus two acts equal seven now distribute the negative two on each term, and that gives us negative 20 plus four x adding up like terms. We get nine X and then adding 20 double size because there's 27 finally divided by nine, we get three now. Substitute this result from step three into the equation, drives and step one. So substituting X equals three into this equation, we get four. Hence, the solution of the system of equations is the ordered pair. Three comma, four 47. Equations Problem 1: dividing both sides of this equation. By six, we get a equals 56 of be that is a is a fraction of B or in other words, we have to multiply be by a fraction to make it a small is a instance a is positive. B is positives. Well, hence be is larger than a and the answer is D. 48. Equations Problem 2: that's merely add the two equations first, aligning him vertically we get and people's Pius two p the cuse cancel an R plus ours to our equals 12 and notice. This is almost the expression we're looking for. All we have to do is divide everything by two and that we have P Plus R is equal to six. Hence the answer is C. 49. Equations Problem 3: it's clear the fresh in the second half of the equation multiply both sides by two. We get to wind minus two is equal to why plus five because the twos cancel, distribute the to we get to while minus four. Subtracting why from both sides at the same time, adding four to both sides gives why this cancels the wise canceled and we get nine. Now go back to the first part of the equation. Namely, that X equals Y minus two. Since we have the value of why is nine, we get nine minus two or seven? 50. Equations Problem 4: first substitute three q plus one and for P. So we get three to the Q plus one over three squared now subtracting that bottom exported from the top. We get combining like terms gives us three to the Q minus one. Now replace que with two r and we get three to the two, are minus one and then just noticed that this is answer choice a. 51. Equations Problem 5: first, let's plug in U equals 18 and U and V equals two into the expression to determine the value of K. So we get 18 minus two over K is equal to eight. Multiply both sides by K we get and then divided by eight we get that K is equal to two plugging that into our expression. Now we're asked for the value of V partly for the value of you. When V is for so plug and four for V adding four to both sides, we get U equals 20. Hence the answer is E. 52. Equations Problem 6: This triple equation contains three equations. The 1st 1 X equals three wide 2nd 1 Why three y equals four z and X equals four Z. Now we're trying to create the expression six acts out of these equations so well, playing this equation here by six we get six X equals six times three wine, which is 18. Why in statement one is true. This eliminates be and see. No one statement to the 20 Z can be written as five times for Z. We did that so we can substitute in X for the four Z. So we get three Why which of course, is equal to acts itself. So we get tax plus five times X, which gives us six X which again is the expression we're looking for in statement to is also true, which eliminates a therefore by process of elimination. The answer is D 53. Equations Problem 7: Plugging the values of P and K into the equation gives us 10 equals X plus Y times three. Now dividing both sides of this equation by three we get X plus y equals 10/3. And to form the average, we divide both sides of this equation by two. So this I will divide by two in the Sybil, multiplied by 1/2 and then cancelling gives us five on the top and three on the bottom. Instead averages 5/3 and answers see? 54. Equations Problem 8: there are many values of X, y, W and Z. Who's some in this equation will be to, for example, of all the terms were one, and we have one plus 1/1, which gives us one plus one or two. And since everything's equal the one this expression will have the same value. Now suppose we choose execute three. Why to be two, WB one and Z to be to. Then they will satisfy this equation, and we can check that real quick. We get three over to plus 1/2 which gives us four house, which is to, and when we plug it into this expression will get a different value. The why is, too the X is three Z is, too, and the W is one getting a common denominator of three. We have to plus 6/3, which is a third's, since we have a different value here we have to, and here we have 8/3. There's not enough information to determine an answer 55. Equations Problem 9: Let's translate this expression into an equation. 4% is point 04 and when you're dealing with percent of makes multiplication, so 0.4 times people askew is equals eight, dividing both sides. By 0.4, we get P plus Q equals eight over 80.4 which is 200. Now. Subtract p from both sides and we get que is equal to 200 minus P. Now we're told that he is a positive energy and this expression will be a large as possible . And P is a smallest possible against his piece of positive manager. The smallest possible value of P is one, and we get 200 minus one or 1 99 Hence the answers D. 56. Equations Problem 10: Let's dig out a factor of X to the fourth from this expression and then replace it with seven y and see what it leads to. Excellent. Fifth can be written as extra fourth times X to the one still equal before now replacing X 1/4 was seven over why we get and were asked to find the value of X in terms of why, in other words, solve this equation for X in terms of why so multiply both sides by y. Why is cancelled? We have seven x is equal to four wire and now survival sides, with seven in the seventies canceled. So we have X equals for why over seven hence, the answer is B. 57. Equations Problem 11: the promise, asking what is a little less in terms of biggest. In other words, we have to solve this equation for little s and notices on both the top and the bottom of the fraction. So we're gonna have multiplied by the LCD to clear the fraction in the LCD is 12 times little s plus big ass, and four will cancel the 12 3 times the little s plus biggest cancels entirely. So we're left on this side of the equation with three times and on the side of the equation , that three goes in 12 4 times, we have four times distributing both of four and the three we get. Subtracting three little s from both sides and subtracting four biggest from both sides. We get little s. My guess is cancelled here. The little s is cancel here, and we have minus seven s on this side, which is answer choice D 58. Equations Problem 12: note that 81 is three. Raise to the fourth power. So this equation tells us that X must equal four, plugging that into the expression we get. Now we glance at the answer choices and notice choices. A and B have a seven. There's no way to get seven out of three and four, so eliminate those and all the remaining answer choices have a 12 minute, so see if we can manipulate this to get a 12 out of it. In order to be able to multiply the three in the four, they must have the same exponents. Now three to seventh can be written as three squared times, three to the fifth power time for the fifth power since they have the same export. Now we can combine them, right. That is three times for to the fifth Power, which gives us three squared times 12 to the fifth. Multiply in the 34 which is nine times 12 to the fifth power, which is answer choice, D 59. Equations Problem 13: Let's translate the claws into an equation we have. P over 19 is equals one less than three times. Q over 19. Those that even though less than was mentioned. First, we subtract with one from the expression three. Q over 19. The less than changes the direction of the subtraction. That's just a quirk of the language. Now multiply both sides of this equation by 19 and we get P equals three. Q. By distributing 19 on each term, it will cancel the 1st 1 minus 19. Now just noticed that is answer choice E. 60. Equations Problem 14: remember that even exponents destroyed negatives, so this the expression could be rewritten as eight to the to end. Now eight KB written is two cubed. So we get to cubed to the to end the power raised to another power. You multiply the powers and we get to to the three times to in or two to the six in. They'll bring in the other side of the equation and noticed now that we have the same base , namely two. Hence the exponents must equal each other. So six in must equal eight plus to end and subtracting to him. We get foreign, dividing by four gives us in equals two and the answer is B. 61. Equations Problem 15: since her expression has a Z in it. Let's solve the second equation for Z will point both sides by two. We get Z equals two. I now plug in Why, over two for X in our expression and plug in to y for Z. And you could write to why over one if you wish toe balance it out. So we get why over to times the bottom flipped over, which is one over to why and cancelling the wise and we'll find the twos. We get the square root of 1/4 which is 1/2 since the answer is D. 62. Equations Problem 16: that said the two equations, as is aligning the equations vertically we get and no adding we get four X Plus four. One equals 12 and now divide out the factor of four to form the expression X plus. Why that we're looking for in the force. Cancel and we get next plus water and 12/4 is three. Instead, answers see. 63. Equations Problem 17: Let's add the two equations first, aligning them vertically. We get seven X minus Y equals 23 and we're gonna add the two equations. So adding the exes, we get seven X minus. Access six X in similarly seven y minus. Why is six wine and 31 plus 23 is 54 now. Divide each term by six and we get X Plus y equals no. Hence the answers E. 64. Equations Problem 18: Let's add the three equations, aligning them vertically. We get X plus y equals for a over five and then Z plus X equals nine a over five. Adding up the excess, we get two of them. Adding up the wise we get to and adding disease. We get Tuas well, and then on the right side, they all have a common denominator of five. So seven plus four plus nine is 20 a over the common denominator five, which reduces to four a cancer in the common factor of five. Now divide both sides of this equation by two that forms the expression that we're looking for, namely X plus y plus Z, and it equals two A. Hence, the answer is C. 65. Averages Definition: the average of in numbers is there some divided by N. 66. Averages Example 1: by the definition of an average, we get the some of the three expressions divided by their number, which is three now, factoring out of three and canceling, we get X Plus two. It's the answer is D. 67. Averages Weighted Average: weighted average. The average between two sets of numbers is closer to the set, with more numbers. For example, if on a test three people answered 90% of the questions correctly and two people answered 80% correctly than the average for the group is not 85 but rather 86 as the calculations show. Here, here, 90 has a weight of three is being multiplied by the three people, and 80 has a weight of only two, so the average is closer to 90 than it is to 80 as we have just calculated. 68. Averages Using an Average to Find a Number: using an average to find a number. Sometimes you will be asked to find a number by using a given average. An example will illustrate. 69. Averages Example 2: Let's look the five numbers B, A, B, C, D and E for me, their average we get. There's some divided by five because there's five terms is negative Now. We're told that the sum of three of the numbers is 16 and it doesn't matter which three we choose. Let's go ahead and choose. The 1st 3 numbers have a sum of 16 that gives us 16 plus de plus e over five equals negative 10. Now let's turn this into an average. That is this form. They manipulate the expression so that D plus C, divided by two, is equal toe number well by both sides. By five we get and subtracting 16 we get minus 66 and finally dividing by two to form the average. We get negative 33. Since the answer is a 70. Averages Average Speed: average speed is equal to the total distance divided by the total time. 71. Averages Example 3: Although the formula for average speed is simple, few people saw these problems correctly because most failed to find both the total distance and the total time. The total distance in this problem is 50 miles for the first part of trip, one hour at 50 MPH and for the second half of the trip, it's 60 MPH for three hours three times 60 which gives us 50 plus 1 80 or 2 30 Now the total time is one hour for the first part and three hours for the second part. So we get the average speed is the total distance divided by the total time, which is 3 30 divided by four. And this reduces to 57 and 1/2 which is answer choice E No. The answer is not the mirror average of 50 and 60. Rather, the average is closer to 60 because he traveled longer 60 MPH three hours than at 50 MPH, one hour 72. Averages Problem 1: for me, the average we get this some of the terms divided by the number of terms which is to is equals 10 adding that the light terms this is one p you know that one is not written in front. It's assumed to be there. And yet five p over two equals 10. Why both sides by two and then dividing both sides by five, we get people's four. 73. Averages Problem 2: we have the 1st 6 consecutive managers whose average is 9.5. So we have the 1st 3 integers less than 9.5 and the 1st 3 integers greater than 9.5. That is, we are dealing with the number 789 10 11 and 12. Now they want to know what is the average of the last three numbers? Well, clearly average between 10 11 and 12 is 11 but let's confirm that adding up the numbers and dividing by three we get 10 plus 11 is 21 plus 12 is 33 and 33 divided by three is 11. Instead, answers D. 74. Averages Problem 3: the average of the consecutive positive integers one through N is there some divided by the number of numbers which is clearly in Now we're told that in denotes the some of the positive injures one through in the top of this fractions precisely that the some of the images went through in. So replace it with us and we immediately get statement one which eliminates B and C. Now solving this equation for ass will multiply both sides by end. Canceling the ends we get s is equal to in a in statement too is false, which eliminates D therefore, by process of elimination, the answer is a 75. Averages Problem 4: the average speed at which car X traveled is the total distance divided by the total time and the total time is 30 minutes. The average speed at which car y traveled is again total distance divided by total time, which is 20 minutes. The two fractions have the same numerator and the denominator for car Y is smaller 20 versus 30. Hence the average MPH at which cart why travel is greater than the average MPH. Which Car X travel. The statement one is false and Statement three is true as the statement to we do not have enough information to calculate the distance between the cities in statement to need not be true and the answer is C. 76. Averages Problem 5: the average of P Q and R. Is there some divided by the number and there are three items, so we divide by three. Now. P plus Q can be replaced with our giving R plus R over three, which reduces to 2/3 or hence the answer is C. 77. Averages Problem 6: often on the test, you'll be given numbers in different units. When this occurs, you must convert the numbers into the same units. This is obnoxious, but it does occur on the tests will be alert to it. In this problem, we must convert 15 minutes into hours. Now there are 60 minutes in an hour, so 15 minutes is equivalent to 1/4 of an hour. Hence the average speed, which is the total distance divided by the total time, is X for the distance. The total distance and the total time is why hours plus 15 minutes, which we converted into 1/4 of an hour. And now just notice that this is answer choice, see? 78. Averages Problem 7: forming the average of the five numbers we get the plus w plus tax plus why plus Z divided by five, we're told, is 6.9 No. One of the numbers deleted. Let's let that be the number Z. So we now have divided by four for me, the new average and we're told that's equal to 4.4. Well, playing both sides before we get 17.6 and plugging that into the original expression appear we get 17.6 plus Z divided by five equals 6.9. Now, solving this equation we get Z equals 16.9, which is answer choice d. 79. Averages Problem 8: let the four numbers B, A, B, C, D and E. Since their averages 20 we get there's some divide by four is 20 now let the number that is real, that is removed b d. Then we'll have a plus B plus C divided by three because there's now only three numbers and we're told that that is equal to 15. Moved by both sides by three and we get 45. Now substitute that in for a plus B plus C and original equation, and we get 45 well played by four and then subtracting 45. We get d equals 35. Hence the answer is deep. 80. Averages Problem 9: let the other number B y that the average of the two numbers X plus y divided by two is pi over two. Now multiply by two To clear the fractions, we get X plus y equals pi. Now they are asking, What is the other number in terms of that? In other words, we saw the equation for why, in terms of X, which gives us pie minus X, which is answer choice, see. 81. Averages Problem 10: This is a weighted average problem. Because more disk were purchased on the second day. Let X be the number of disc purchased on the first day. Then, since each disk costs 50 cents, get 50 times X equals the total cost, which is 25. Divide both sides by 25 0 and we get X equals 50. That would be the number this purchased on the second day, then 0.30 times. Why is equal to 45? So why is equal to 1 50 now forming the weighted average we get? The average cost is the total cost divided by the total number, and the total cost is 25 plus 45 and the total number is 50 which we derived here, plus 1 50 which we drive here. And this gives us 70 over 200 which reduces the point 35 which is enter choice. See 82. Ratio Introduction: ratio ratio is simply a fraction. The following notations all expressed the ratio of extra Why Ex school and why X divided by why an X forward slash y writing to numbers as a racial provides a convenient way to compare their sizes. For example, since three divided by pi is less than one, we know that three is less than pie. Racial compares to numbers. Just as you cannot compare apples and oranges, so too, must the numbers you are comparing have the same units. For example, you cannot form the ratio of two feet to four yards. Because the two numbers are expressed in different units feet versus yards. It is quite common for the test to ask for the ratio of two numbers that are expressed in different units before you form any ratio. Make sure the two numbers are expressed in the same units 83. Ratio Example: the ratio cannot be formed until the numbers are expressed in the same units we have here. Feet versus yards. Let's turn the yards into feed. There are three feet in the yard, so four yards is equal to four times three feet or 12 feet. Now we conform the ratio, which is two feet to 12 feet, which gives US 16 or in ratio notation one cool in six. Hence the answer is D. 84. Proportion Introduction: proportion proportion is simply an inequality between two ratios or fractions. For example, the ratio of extra Y is equal to the ratio of 3 to 2 is translated as follows or in ratio notation. We can read it as thus. 85. Proportion Direct Proportion 1: two variables are directly proportional. If one is a constant multiple of the other. In this case, why equals K axe were K is a constant. The above equation shows that as X increases or decreases, so does why this simple concept has numerous applications and mathematics, for example, in constant velocity problems that distance is directly proportional to time. D equals velocity times time where the is a constant no. Sometimes the word directly is suppressed. So instead of saying why is directly proportional to acts, we just say, Why is proportional to X? 86. Proportion Direct Proportion 2: in many where problems as one quantity increases or decreases, another quantity also increases or decreases. This type of problem can be solved by setting up a direct proportion. 87. Proportion Example 1: translating the ratio of wide two X is equal to three goes. Why over X equals three and the sum of why and axes 80 translates into y plus acts equals 80. No solving this equation for why we get why equals three X and substitute this into the other equation. Adding life terms we get four X equals 80 divided by four. We get X equals 20. Employing this into the other equation we get. Why equals three times 20 which is 60 hence the answer is E. 88. Proportion Example 2: As time increases, so does the number of surfboard shape. Hence, we can set up a direct proportion first convert five hours into minutes since their 60 minutes in an hour, we get five times 60 which is 300 minutes now. Let X be the number of surfboards shaped in five hours, then forming the ratio. We get three surfboards in 50 minutes versus some unknown number of surfboards. In 300 minutes, we'll find both sides of this equation by 300. We get and reducing. We can cancel a zero here, and that gives us 90 over five, which reduces to 18. Hence dancers see. 89. Proportion Example 3: As the distance on the map increases, so does the actual distance. Since we set up a direct proportion, let X be the actual distance between the sister cities forming the proportion yields we have one edge represents 150 miles and that the city's air 3/2 inches apart. So we get three and 1/2 is to the actual distance X and cross small plane. We will get X equals three and 1/2 times 1 50 and this reduces to 500 in 25. Hence, the answer is D. 90. Proportion Order: No. You need not worry about how you form the direct proportions so long as the orders the same on both sides of the equal sign the proportion the previous example could have been written as inches two inches as miles are two miles. In this case, the order is interest inches and miles, two miles. However, the following is not a direct proportion, because the order is not the same on both sides of the equal sign. On the left, we have inches two miles and on the right we have miles, two inches. 91. Proportion Inverse Proportion 1: one quantity increases while another quantity decreases. The quantities are said to be inversely proportional. The statement Why is inversely proportional to X is written as follows. Why equals K over acts where K is a constant multiplying both sides of this equation by X. We'll cancel the exes and we'll get X Times y equals K, hence, in an inverse proportion, the product. Why times acts of the two quantities is constant because they're set equal to a constant K . Therefore, instead of setting ratios equal, we set products equal. 92. Proportion Inverse Proportion 2: in many where problems as one quantity increases, another quantity decreases. This type of problem can be solved by setting up a product of terms. 93. Proportion Example 4: as the number of workers increases, the amount of time required to assemble the car decreases. Hence, we accept products of the terms equal that XP the time it takes 12 workers to assemble the car. For me, the equation yields seven times eight equals 12 times x, which gives US 56 equals 12 times acts divided by 12. We get X equals 56/12 which can be reduced to four and 2/3. It's the answer is C. 94. Proportion Summary : to summarize if one quantity increases as another quantity also increases set ratios equal if one quantity increases as another quantity decreases, set products equal. The concept of a portion can be generalized to three or more ratios. A, B and C are in the ratio 345 beings A is to be as three is the four, which is what this equation is saying and a is the sea, as three years to five knows the order is the same and be is to see as four is 25 95. Proportion Example 5: since the angle. Some of a triangle is 180 we get a plus B plus C equals 1 80 now forming two of the ratios up here we get A is to be as 5 to 12 and we also get a is to see as five is to 13. Solving the first equation for B we get be equals 12 5th and solving the second equation for See We Get C equals 13th ifs of a plugging these values into the equation we get and simplifying this that I didn't have all these fractions will get six a equals 1 80 and divided by six. We get a equals 30 hence, the answer is C. 96. Ratio and Proportion Problem 1 : first convert all the units into inches. There are 12 inches in a foot, so two feet and three inches is 27 inches and their suit 36 inches in a yard. So two yards 72 inches Now. For me, the ratio we get 27 over 72 which reduces that 3/8. Hence the answer is C. 97. Ratio and Proportion Problem 2: let X and Y did note the numbers. Since the ratio of the two numbers is 10 we get X over one equals 10 and their differences 18 So x minus. Why equals 18 Solving this equation for acts we get X equals 10. Why? Substituting into the other equation we get 10 y minus y is equal 18 or nine y equals 18. Why equals two plugging this value for why, into the other equation we get X over two is equal 10 Well played by two. We get X equals 20 and we're looking for the smaller number. Hence the answer is to 98. Ratio and Proportion Problem 3: let X and Y denote the angles of the triangle and let X be the base Ingles. Since the ratio is 1 to 3, we get X over wide equals 1/3. Further, since the English some of the triangle is 180 degrees. We get X plus X plus. Why equals 1 80 or two x plus? Why equals 1 80? Solving this equation for why we get why equals three X substituting this value? Why into the other equation we get to explain plus three x equals 1 80 or five X equals 1 80 So x equals 36. Plugging that back into the equation. Up here we get three times 36 or 108 It's the answer is E. 99. Ratio and Proportion Problem 4: This is a direct proportion. As the distance increases, the gallons of fuel consumed also increases sitting ratios equal. We get 80 gallons is to 320 miles as some unknown number of gallons is to 700 miles. Well played both sides by 700 we get, and this reduces toe 1 75 hence the answer is E. 100. Ratio and Proportion Problem 5: This is an inverse proportion as the number of boys increases, the time required to complete the job decreases. So we set products equal. Now, two hours and 30 minutes is 2.5 hours. Times two boys will be the same as when three other boys joined. The two that are there will give us five times t. So multiply in the left side. We get five equals five t divided by five. We get teak was one hence, the answer is a 101. Ratio and Proportion Problem 6: This is a direct proportion. As the amount of flour increases, so must the amount of shortening. To get anything the same units list convert the pounds into ounces. There are 16 ounces in a pound, so half of a pound will be eight ounces, forming the ratio we get. Eight ounces of shortening is the 14 ounces of flour, as some unknown quantity of shortening X is 2 21 ounces of flour. We'll find both sides of this equation by 21 and reducing we get 12 hence, the answer is D. 102. Ratio and Proportion Problem 7: Most students struggle with this type of problem, and the G R E considers them to be difficult. However, if you can identify whether problems a direct proportion or an inverse proportion that it is not so challenging in this problem. As the number of widgets increases, so does the absolute cost. This is a direct proportion, and therefore we set ratios equal. So W regions is to de dollars as 2000 widgets is to an unknown cost Now cross multiply. In this equation we get X w equals 2000 times D, dividing both sides of this equation by W. The W's cancel. We get X equals 2000 D over W, which is answer choice, see. 103. Ratio and Proportion Problem 8: begin by adding the two equations as is that would cancel the wise and the one in negative ones. So we have four ax minus nine. Z equals zero or four. X is equal to nine Z, and we're trying to form the ratio of ecstasy. So divide both sides by Z reminds, wielded by both sides by four. At the same time, cancers of fours and gives his ex oversee is equal to nine force after cancelling disease, and this is answer choice E. 104. Ratio and Proportion Problem 9: this is a direct proportion. As the time increases, so does the number of steps that the spinner takes. Setting ratios equal. We get 30 steps is to nine seconds as some unknown number of steps is to 54 seconds. We'll find both sides by 54 and this expression reduces to 1 80 hence dancers D. 105. Ratio and Proportion Problem 10: we are asked for the ratio of X two y. So let's divide this equation by why the form the ratio of X over Y. I mean my most divide by five at the same time. So here the fives cancel and we get X over y. And here the wise cancel when we get 6/5. Hence the ratio of extra Why is 6 to 5 or in racial notation, it's six. Colin five Instead answers d. 106. Exponents Introduction: exponents. Exponents afford a convenient way of expressing long products of the same number. The expression B to the end is called the power, and it stands for B times. B times be dot, dot dot times be Where there are in factors of B B is called the base and it is called the exponents Be to the zero is, by definition equal toe one. 107. Exponents The Rules 1: There are six main rules that govern the behavior of exponents. If you multiply the basis of two powers, you add the exponents and caution. You cannot add exponents unless the powers and the bases are exactly the same. If you haven't X cubed, plus the next to the fourth, you cannot Adam because they are not like terms. But if you haven't X cubed plus another X cubed, then you can Adam and you have one here and one here and you get two of them. Two x cubed and a power raised to a power is equal to the product of the powers, and you could distribute a power over a product. X Times Why gives you extra A times Why today and be careful. This is not true for some or difference X plus y to the A. Power does not equal X to the A Plus what? Why do they? And that's also true for a difference quotient. You could just distribute the exploding over each term as rule for shows 108. Exponents The Rules 2: for a quotient of two powers you subtract the exponents at the expert on the top is larger than the result is on the top. If the exponent on the bottom is larger than the result is on the bottom and a power read and a base raise to a negative power will reciprocate the base. So Z to the dagger three is simply one over Z Cube, cautioning negative exponents does not make the number negative. It merely indicates that the base should be reciprocated. So three raised the negative two does not equal negative 1/3 squared. In fact, three raised to the negative to power simply reciprocates the three and you get 1/3 square , which, of course, is 1/9. Problems involving these six rules are common on the test, and they are often listed as hard problems. However, the process of solving these problems is quite mechanical. Simply apply the six rules until they can no longer be applied 109. Exponents Example 1: here we have a power raised to a power. So we multiply the exponents and we get X Times X to the 10th Power. Now there's a one here is in the exponents, even though it's not written. And since the base is the same, we add the exponents X to the 11th and a power overpower you subtract exponents get next to the 11 minus for which gives his ex to the seventh. Hence the answer is C. 110. Exponents Example 2: canceling the common factors of three. We get one third time's 1/3 times, 1/3 times 1/3 which is 1/3 raised to the fourth power. Hence, the answer is a. 111. Exponents Example 3: first factor, the six into two times three. Now apply the exponents each term individually. And we have to to the fourth times three to the fourth over three squared. Since we have the same base here, we can subtract the exponents and we have 3 to 4 minus two, which gives us three square. Hence the answer is D. 112. Roots 1: roots. The symbol is read the in through of B, where in is called the index B is called a base, and this symbol is called the radical Thean. Through of B denotes that number, which raised to the power heels be in other words, a is the in through of B. If a to the end equals B, for example, the square root of nine is equal to three because three squared equals nine and the Cuba of Negative eight is negative, too, because negative two cubed is equal to negative eight. Even roots occur in pairs, both a positive route and a negative root. For example, the four through 16 is too, since two to the fourth power 16. But the four through 2 16 is also equal to negative two. Since negative to to the fourth. Power is also 16. Odd routes occur alone and have the same sign as the base. For example, the cube root of negative 27 is negative. Three. Since negative three Cube is negative. 27. If you are given an even route, you're to assume that it is the positive route. However, if you introduce even routes by solving equation then you must consider both positive and negative roots. For example, if you're given the equation, X squared equals nine and then take the square root of both sides. You get plus and minus the square root of nine or X equals plus and minus three. 113. Roots 2: square roots and cube roots can be simplified by removing perfect squares and perfect cubes , respectively. For example, eight can be factored into four times two and four is a perfect square, which gives us to root. Two. Note. This step here is usually skipped, and we go directly from pulling out the perfect square to from the four. Likewise, 54 can be factored into 27 times two and 27 is a perfect cube, namely three cube, hence the cube root 27 is three. 114. Roots 3: radicals are often written with fractional exponents. The expression in through to be can be written as B to the one over end. This can be generalized as follows be to the M over end gives us the in through of be another words. The end is the index of the radical and I m is the exponents. Usually this form here is preferred because the number inside the radical is smaller than in this form. For example, 27 to 2/3 power. The three becomes the index of the radical and two becomes the power. Now the Cuban 27 is three and three scored his nine. Using this form for this problem would be much harder in this case, because 27 to the 2/3 again would be the cube root of 27 square, which is the cube root of 7 29 which gives the same answer. But most people will know what the Cuban 27 is that very few people will know what the Cuba does. 729 is 115. Roots 4: if Innis even than in through of X to the power is the absolute value of acts. For example, the fourth root of negative two to the fourth power is the absolute value of native to which is to no mechanically. What's going on here is this. Even exponents is destroying the negative. So negative to to the fourth. Power is positive. 16 and the fourth through to 16 is positive, too, with odd routes that value is not needed. For example, the cube root of negative to cube. It's a Cuban of negative eight because they give to cube is negative eight and cube roots, all on routes preserve negative numbers, so we get back negative, too. 116. Roots 5: to solve radical equations. Just apply the rules of exponents to undo the radicals. For example, to solve the radical equation X to the 2/3 equals four, we can cue both sides to eliminate the radical now a power to a power you multiply. The exponents, which will cancel the threes and give us X squared equals 64 which is four cube. Then take the square with both sides. We get the absolute value of X equals eight because the square root of 64 is eight and dropping the absolute value we get plus and minus eight. No, it's this step here is usually skipped and we go directly from taking the square root of both sides and just writing down plus and minus the result. 117. Roots 6: There are only two rules for routes that you need to know for the test, namely that the root of a product is a product of roots, and the root of a question is a question of roots. Caution. The root of X plus y does not equal the root of X plus the root of why, for example, the square root of X Plus five does not equal. Describe it of X plus the square with five also the square root of X squared plus y squared does not equal X plus. Why this common mistake occurs because it is similar to the falling valid property. The square root of the quantity X plus y squared does equal X plus y. But notice here the terms individually r squared and here is their whole son that is square . If X plus, why can be negative that it must be written with the absolute value. Simple note. In the valid formula, it's the whole term X plus why that is squared, not the individual X and Y 118. Roots 7: to add to roots, both the index and the base must be the same. For example, the Cuban A two plus the four through two cannot be added because the indices, the three in the four are different. Nor can square root of two plus square root of three be added because now the bases two and three are different. However, the cube root of two plus that Cuba to can be added and we get two of them to Cuba, it's of two. In this case, the roots can be added because both the indices, the cube roots and the basis the twos are the same. Sometimes radicals with different bases can actually be added once they have been simplified toe look alike. For example, the square root of 28 plus the square to 27 we can write 28 as four times seven and then pull out the four radical four and radicals four is equal to two, and now we have to radical sevens here and one radical 70. Even though it's not written, there is a one here for a total of three radical sevens 119. Roots 8: you need to know the approximations of the following routes. Radical to is approximately 1.4. Radical three is approximately 1.7 and Radical five is approximately 2.2. 120. Roots Example 1: taking the cube root of the bottom equation. We get that Why is equal to negative two and taking the square root of the top equation? We get X equals plus and minus two. Now, if X equals positive too than X will be greater than why. Because why is negative two? But if X equals negative, too, then X will equal why. Because again, why is negative two? His choice d is not necessarily true. 121. Roots Example 2: translating the expression into an equation we get. Why is equals five Mawr Warming's edition than the square of X? Now it's asking this to solve the falling for X. In terms of wise, we have to solve this equation for X in terms of wine, subtracting five from both sides, we get why minus five equals X squared. Now take the square root herbal sides and we get plus and minus. When we take the square root equals X. Now we're told that X is less than zero. So we take the negative root and now just notice that this is answer choice be 122. Roots Example 3: translating the expression into an equation we get. Why is equals five Mawr Warming's edition than the square of X. Now it's asking us to solve the falling for X. In terms of wise, we have to solve this equation for X in terms of wine, subtracting five from both sides, we get why minus five equals X squared. Now take the square root herbal sides and we get plus and minus. When we take the square root equals X. Now we're told that X is less than zero. So we take the negative root and now just notice that this is answer choice be 123. Roots Rationalizing: rationalizing. A fraction is not considered simplified until all the radicals have been removed from the denominator, the bottom part of the fraction. If a denominator contains a single term with the square root, it can be rationalized by multiplying both the numerator and the denominator by that square root. If the denominator contains square roots, separated by a plus or minus symbol that multiply both the numerator and the denominator by the conjugal, which is formed by merely changing the sign between the roots. 124. Roots Example 4: example, rationalize the fraction to over three. Radical five. We multiply top and bottom of the fraction by the radical five. Excuse is this right here and radical? Five times Radical five is radical 25 the square root of 25 is five. Which gives is this as our answer? Usually the step here is skipped, and we go directly from radical five times Radical five equals five. 125. Roots Example 5: example. Rationalize a fraction to over three minus radical. Five. Multiply top and bottom of the fraction by three, plus radical five, which is the conjugal, which is merely changing the sign of the middle term. In this case, we changed it to a positive because it's negative, but if it were positive, we would change it to a negative, performing the product and foiling the terms on the bottom. We get this extended product and you can cancel these terms. And in three squared is nine and radical. Five Squared is five Now. If you notice that congregates are always a difference of squares, you can bypass this long process here and get three squared is nine minus always a minus. Radical Phi Squared, which is five and nine minus five, is four and cancelling the to we get our final answer. 126. Exponents and Roots Problem 1: start by factoring 20. It can be written as two times two times five and race eighth power or two square times five raise the eighth power And now distributing the the expert on each factor we get two squared to the eighth times, five days and a power to a power. You multiply the power. So to get to to the 16th times five to eighth. Well, this expression to the 16th represents all powers of two of the form to the end. Hence the answer is D. 127. Exponents and Roots Problem 2: first, let's distribute the exploding on the top and bottom of the fraction that gives us two y cubed to the fourth power over X squared to the fourth power. Now distribute the export it on the top We get to to the fourth. Why cube to fourth now a power toe power. We multiply the exponents and to the fourth is 16. So we get wide to the 12 and likewise at the bottom we moved by the two. In the four, we get X to the eighth and when you're dividing bases, use a practice exponents. So we get 16 to the 12th 16 wide to the 12th power times X to the 10 minus eight, which is to therefore the answer is choice a. 128. Exponents and Roots Problem 3: performing the operations. Inside the parentheses we get 31 minus six is 25 and 16 plus nine is also 25 and now you can break up the radicals as radical 25 times Radical 25 which is five times five for 25. Hence the answer is C. 129. Exponents and Roots Problem 4: begin by factoring 55 in the top of the of the fraction. And 55 could be factored into five times 11. Now apply That exploded to each factor. And we get five to the fifth times, 11 to the fifth, over five to the 55th power. And now, sir, practice exponents here and we get 11 to the fifth over five to the 55th minus five, which is 11 to the fifth. Power over five to the 50th power, which is choice, see. 130. Exponents and Roots Problem 5: plugging in the 1/9 we get now. The square root of 1/9 is 1/3 and one night squared is 1/81 get a common dominator but multiplying top and bottom of 1/3 by 27. That gives us 27 minus 1/81 which is 26/81. Hence the answer is C. 131. Exponents and Roots Problem 6: We have a power to a power here. So we multiply the exports which gives nine to the three X now replaced nine with three squared and still raised to three X. And once again, we have a power to a powers. We multiply the exponents and you get three to the two times three X, which is three to the six X hence the answer is C. 132. Exponents and Roots Problem 7: plugging four into the expression we get no give to to the to times square root of four plus two square to four is too. So we have negative to to the fourth power plus two and negative to the fourth. Power is negative. 16. Not positive 16. It would have to be in case in parentheses to be pause of 16. Negative two squared. The quantity squared would be positive 16 60 miles to its negative 14. So the answer is a. 133. Exponents and Roots Problem 8: factoring the numerator of the fraction we get five plus X times five plus X, which can be rewritten as the square root of five plus X. The quantity squared over to and now distribute the radical on the top of the bottom and on topic cancels. Then we get five plus X over radical, too, but that is not offered as an answer choice. So let's rationalise the expression by multiplying top and bottom by radical, too. That would give us two on the top. That will give us two on the bottom and five plus X times radical two on the top, which is answer choice, see. 134. Exponents and Roots Problem 9: this rationalized the expression by multiplying top and bottom by the Contra git, which is two plus radical, five on the bottom. That gives us a difference of squares, which will be two squared, minus radical five square and on top We get foiling it. We would get four. The other two give us to Route five, the inner to also give us to Route five. So we get four root five in the last two radical five times Radical five is five and adding up like terms. On the top we get nine plus four, radical five and the bottom. We have four minus five, which is negative one and bring the negative. Upstairs we get negative nine plus four radical five and then distribute the negative. We get negative nine minus for radical five, which is answer choice a 135. Exponents and Roots Problem 10: by the definition of multiplication, we get four times to to the 12th power now rewriting four as two squared, we have to square times to the 12th. Since the basis of the same. We add the exponents, which gives us two to the 14th power. Instead, answers be. 136. Exponents and Roots Problem 11: working from the innermost privacy out we distribute the exponents on each term. So we get X squared cubed Why cube z all over x y z Now, with the power to a power we built by the exponents And now subtracting one from the sixth and the one from the why and we get X to the fifth. Why square and Z over Z the quantity cube, Cancel disease and now distribute the cube on each term. So we get X to the Fifth Cube. Why squared cubed the power to power remote by the exponents. We get extra 15th. Why? To the sixth, which is answer choice E. 137. Factoring Introduction: to factor in algebraic expressions to rewrite it as a product of two or more expressions called factors. In general, any expression on the test that can be factored should be factored, and any expression that Kenny Unf acted multiplied out should be unf acted. 138. Factoring Distributive Rule: distributive rule. The most basic type of factory involves the distributive rule, also known as factoring out a common factor here. We have a common factor of a here in here, which we pull out in front, and the X and the Y remain when this rule is applied from left to right is called factoring . When the rules applied from right to left. It's called distributing. For example, Here we have three H plus three K that threes air common so you can pull it out in front and you're left with the H in the K. And here we have a five x Y plus 45 x, the X and the five or common. To show that explicitly that there is the five is common. We could rewrite 45 as nine times five and then pull out the five. Actually, you're left with a why and a nine. The distributor rule can be generalized to any number of terms. For three terms, it looks like a X plus a Y plus a Z. We have a common factor of a in all three terms pulling out, pulling that out in front. We're left with a Times X Y Z 139. Factoring Example 1: Let's combine the terms in the given expression, keeping an eye out for the expression X minus y, if it occurs that will immediately replace it with nine. Distribute the negative on each term we get the positive. Why will be coming? Negative and the negative X over three will become a positive in the multiply top and bottom by three to get a common denominator and likewise for why combining the light turns we get 4/3 axe and negative 4/3 one factory out the common factor of 4/3 were left with X minus y. And that's the term we're looking for. So replace it with nine. Cancel the threes and we get 12 hence, the answer is D. 140. Factoring Example 2: we have a common factor of two to the 19th power in the numerator. So factoring that out, we had 20 twos. Here we removed 19 of them, so one remains. And here we're pulling out all 19 of the twos. But something has to remain. That's the number one and to minus one is one. So we get to to the 19 over to to the 11th. Subtracting the exponents we get to to the 19th minus 11 which is to do they hence. The answer is C. 141. Factoring Difference of Squares: difference of squares. One of the most important formulas on the test is the difference of squares formula. Caution. A sum of squares X squared plus y squared does not factor. 142. Factoring Example 3: first factor out the common factor of eight in the numerator and the common factor of foreign in the denominator. No X squared minus four is the difference of squares because four Kimmy written is to square reminds will cancel the four into the eight writing that is a difference of squares . We get X Plus two and X minus two, and now just cancel the X Plus two factor that leases with two times X minus two. I just noticed that is answer choice a. 143. Factoring Perfect Square Trinomials: perfect square. Try no meals like the difference of squares formula. Perfect score. Try no meal formulas are very common on the test. For example, X square plus six X plus nine is a perfect square. Try no meal because the middle term is twice the square root of the outer to two terms, which is shown here explicitly. So it factors into the square root of the first term and the square root of the last term, the quantity square. 144. Factoring Example 4: notice that the middle term is twice the square root of the outer two terms. Since this is a perfect square, try no meal and we could write it as our minus s. The quantity squared is equal to four. Now, taking the square root of both sides of this equation, we get our minus s equals plus and minus to substituting that into our formula. We get to raise this six power and it is plus and minus, but the even exploded will destroy the negative. So we get a positive regard result, regardless of whether we plug in positive two or negative two. And to the six. Power is 64. Hence, the answer is E. 145. Factoring General Trinomials: general, try no meals. The expression tells us that we need to numbers whose product is the last term and who some is the coefficient of the middle term. Consider the try no Mill X squared plus five X plus six. Now two factors of six or one in six but one plus six does not equal five. The middle term, however, two and three are also factors of six and two plus three equals five Hence X Square plus five X plus six factors into the quantity X plus two times the quantity X plus three. 146. Factoring Example 5: two and negative nine are factors of 18 and negative. Nine plus two is equal to native seven. The middle term it's the equation factors into an X minus nine in an X Plus two setting. Each factory called zero we get X minus nine equals zero or X plus two equals zero, which gives us X equals nine and X equals negative, too. Hence the answer is D. 147. Factoring Complete Factoring: complete factory When factoring in expression. First check for a common factor, then check for a difference of squares. Then, for a perfect square, try no meal and then for a general, try no meal as an example. This factor this expression. First check for a common factor two exes in each term. Factor in the two x out of each term. Yields two x times X squared minus X minus six. Next, there is no difference of squares and X squared minus X minus six is not a perfect square. Try no meals, since the middle term acts does not equal twice the product of the square root of the first term in the square root of the last turn. Now negative three and two are factors of six. Who's some is negative one, which is the coefficient of the middle term. Hence the expression factors into X minus three times X plus two 148. Factoring Problem 1: Let's see if we can manipulate this expression to dig out the given expression notice. Here we have a difference. So let's subtract seven X from both sides to create a difference in this expression. So we have three wind minus seven. X equals negative five. Now we have a common factor of seven between the 2149. Pulling that out, we get three y minus seven x, which is precisely the expression we're looking for substituting in negative five. We get negative 35 which is answer choice be 149. Factoring Problem 2: Let's see if we can dig out the expression X minus minus. Why out of this extra given expression? No, see that we have a common factor of two. We pull that out and we're left with notice that this expression is a perfect square. Try no meal because the middle term is twice the square root of the outer two terms. So we could write it as a perfect square, namely X minus. Why the Quantity square? And this is precisely the term that we're looking for. So replaced the X minus y with P and we get to Peace Square. Hence the answers E. 150. Factoring Problem 3: we were asked to solve four p in terms of Q, so notice that we have a P here and Alsop embedded in the radical. So we're gonna square both sides to open up the radical. So we get peace. Where'd And on this side the radical cancels because we squared it now, bring everything to the left side. So we get peace squared minus two p. Q. And then this term will become plus Q squared equal to zero. And now notice that this is a perfect square. Try no meal. So we converted as p minus que the quantity squared equals zero. Take the square root herbal sides. We get P minus. Que is equal to zero. At incurable sides, we get P equals Q Hence the answer is a 151. Factoring Problem 4: Let's multiply both sides of the equation by five to clear the fraction we can't solidifies and we get X squared equals one times five or five. Now subtract five from both sides. Now we need to factor this equation North's that 15 is five times three can be factored into five times three and five minus three is the middle term, too. So it factors into an accident x of five and a three. And since the middle term is positive, the larger product, which is a five in the axe, takes the positive, and the smaller product, which is the three in the axe, takes the negative, setting each factor equal to zero. We get X equals negative five and X equals three. Now three is not offered as an answer choice, but negative five is, which is answer choice. A 152. Factoring Problem 5: since we were asked the question about the difference of the roots of an equation. We have to solve the equation. Start by dividing every term by five, because there is a common factor of five, and that gives us why Squared minus four y plus three is equal to zero. The three factors into three times one and three plus one is four, which is the middle term. So this factors into a why in a why a three and a one and a negative and a negative setting each factor equal to zero we get Why equals three and why equals one. They were asked to form the difference of the roots and then double it. So three minus one is to and doubling that we get four now the absolute value for is for and also we checked the opposite order one minus three gives us a day of to and doubling that we get negative four. And the absolute value of negative for is also for Henson. Both cases we get four, and the answer is D 153. Factoring Problem 6: start by factoring out the common factor of seven. In the numerator we get X squared plus four X plus four. No notice. The numerator is a perfect square. Try no meal. So it factors into X plus two the quantity squared. And since we have the same expression on the top and bottom, we can cancel it and we're left with seven. Hence, the answer is a 154. Factoring Problem 7: this factor at a common factor of seven to the eighth power in the numerator since the night Factors of 71 remains because we pulled out eight of them and something has to remain here. When we pull out all see all eight factors that will be a one, which gives us 78th times, 8/8. It's cancel and you get seven to the eighth power, which is answer choice D, and you might be surprised learned that this would be considered a very hard problem, not because it's inherently difficult, but because most students would miss it. 155. Factoring Problem 8: Let's factor the expression looking for the for these terms to appear. But we have a difference of squares here, which factors into X plus y and X minus y. And sure enough, the given expressions appeared immediately. So substituting in their values we get five. We get 10 times five, which is 50 against the answers. A. This problem can also be solved by adding the two equations. However, that approach will lead to a long, messy fractions. Writers of the test put questions like this one on the test to see whether you will discover the shortcut. The premise being that those students who do not see the shortcut will take longer to solve the problem and therefore will have less time to finish the test. 156. Factoring Problem 9: notice that X minus Y is a common factor. So factoring that out in front, we get X minus y. So it's gone from here and this ex remains and it's gone from here and this z remains. And now notice that this is answer choice. See now also, to make this a little clearer, we could say Let w equal X minus y And then the expression becomes X times w Just replacing X minus y with W minuses e times w now is visually a little simpler, and we clearly have a common factor of w factoring that out in front. We're left with an X here in a Z here and now just replace the W. It's just a temporary substitution. Replace it with X minus y and again we get answer choice, see. 157. Factoring Problem 10: the left side of the equation is a perfect square. Try no meal, so we get X squared minus twice. The product of X and y plus y squared is equal to the right side, which is X squared. Plus why square now Subtracting X square from both sides. And why square from both sides? We get minus two. X Y is equal to zero and dividing by negative too. We get X Y equals zero hint. Statement three is true, which eliminates choices. A be and see now statement, too, is false. For example, zero times five is equal to zero, and in this case, the why has a value of five, not zero, and a similar analysis shows that statement. One is also false. Hence the answer is D. 158. Factoring Problem 11: first, let's factor the difference of squares. We get X plus y times X minus y. Now both excel wire prime numbers, and they're both bigger than two. Hence, both of them must be odd numbers because two is the only even prime. Now recall that if you add two odd numbers, you get an even number. For example, three plus 58 And likewise, if you multiple. If you subtract two odd numbers, you get an even number. Since both of these numbers air, even they contain a factor of four. Hence, the expression must be divisible by four. And the answer is B. It's also show this explicitly x plus y equal to P because it's even an X minus y equal to kill, because it's even as well, since we get four p times. Q. Since we've written the expression as a multiple off four, it is divisible by four and again, the answer is B 159. Factoring Problem 12: Let's try to dig out X plus y over X minus. Why? Out of this expression? When we find it, we'll replace it with 1/2 factory down an X in the top and on the bottom. Now cancel the exes and we're left with. Now, this is almost what we want. But the order on the the NA Meter is wrong. So factor out and negative. And this right, the negative in front, the whole fraction and change order in the bottom, writing the positive X first and then the negative. Why now? This is exactly the expression we wanted. So we can replace it with 1/2. And we noticed that the answer is B. 160. Algebraic Expressions Introduction: a mathematically expression that contains the variables called an algebraic expression. Some examples of algebraic expressions are X square three X minus two y etcetera to algebraic expressions air called like terms. If both the variable parts and the experts are identical, that is only part of the expressions that can differ are the coefficients. For example, five Y cubed and three has White Cube are like terms, as our X Plus y squared ends negative seven times X plus y squared. However, X cubed and White Cube are not like turn because of variable parts are different, nor are X minus y and two minus y. 161. Algebraic Expressions Adding and Subtracting: adding this attracting algebraic expressions on Lee light turns may be added or subtracted to add or subtract light turns. Murli ADDers of Practical Efficient. For example, here we have one X squared, even though the one is not written, plus three X squared, which gives us a total of four x squared or too radical X minus five. Radical X gives us to minus five or negative three radical acts. 162. Algebraic Expressions Commutative Property: you may add or multiply algebraic expressions in any order. This is called the community Property, so reading from left to right X plus Y is the same as Y plus X and X. Times Y is the same as white times acts. For example, negative two X plus five x can be rewritten with the five x first and the negative two x second and five minutes to us three. So we get three x caution. The community of property does not apply to division or subtraction. Here, too is equal to six divided by three. If we change the order to three divided by six, we get 1/2 so they're not equal and negative one is equal to minus three, and that does not equal three minus two because that's equal deposit of one. 163. Algebraic ExpressionsAssociative Property 1: when adding or multiplying algebraic expressions, you may regroup the terms. This is called the Associative Property Nurse. For addition, we have on both sides of equal sine X plus y plus Z in that order X plus y plus Z. But on the left side of the equation, the 2nd 2 terms are grouped together and on the right side of the equation. The 1st 2 terms are grouped together most of these formulas that the variables have not been moved on. Lee, The way they are group has changed on the left side of the formulas. The last two variables are grouped together on the right side of the formulas. The 1st 2 variables are grouped together. 164. Algebraic ExpressionsAssociative Property 2: Here are two more examples for the associative property here in here. The associate property doesn't apply to division or subtraction as this example here. And this one shows notice in the first example appear that we change the subtraction into a negative edition. X minus two X was rewritten as X plus the negative of two x This allowed us supply the associative property over edition. 165. Algebraic Expressions Parentheses: parentheses. When simplifying Al just break expresses with nested parentheses, work from the innermost parentheses. Out here we have two sets of parentheses, and the innermost are right here. Now two X minus three X is negative acts and a negative times a negative gives us a positive, so we have y plus acts. Now these parentheses have no mathematical purpose any longer, so we can simply add the five acts and the axe and get six x plus y. Sometimes when an expression balls several pairs of parentheses, what are more pairs are written as brackets. This makes the expression easier to read as shown below. 166. Algebraic Expressions Order of Operations: order of operations. Pim Dos, when simplifying algebraic expressions, perform operations within parentheses first and then exponents and then multiplication and then division and then addition. And lastly, subtraction. This can Maybe this can be remembered by the demonic pin Does. Please excuse my dear Aunt Sally, this pneumonic isn't quite precise enough, though Multiplication division are actually tied in order of operation, as is the pair addition and subtraction When multiplication and division or addition its attraction appear at the same level in an expression. Performed the operations from left right, For example, six divided by two times four. The division and the multiplication are in the same rank. And since the division comes first, we do it first. And we can emphasize that by placing it inside parentheses and perform the operation. We get three times four or 12 to emphasize this left right order. We can use parentheses in the demonic Pim dos as follows 167. Algebraic Expressions Example 1: working from the innermost presidency. We performed this division before the addition and we get to plus one, which is three. No, bring in the other terms. Now we apply this exponents before reply the multiplication and we get five minus 27 times three and we do this multiplication before the subtraction. Now we have five minus 81 which is negative. 76. Now let's bring in the other two in a negative times. A negative is a positive. So two plus 76 gives us 78. Hence the answers E. 168. Algebraic Expressions Foil Multiplication 1: foil multiplication. You may recall from algebra that when multiplying two expressions, you use the foil method, which is an acronym for First Outer Inner. Last has shown in this diagram simple. Find the right side we get X squared plus two x y plus y squared. If the product was two negatives instead of two positives, the only thing that would change is this positive would become a negative. These types approx occur often, so it is worthwhile to memorize the formulas. Never, lest you should still learn them a foil method of multiplying because of forms do not apply in all cases. 169. Algebraic Expressions Foil Multiplication 2: examples. Foil filling the 1st 2 terms. The two in the X. We get to acts than O for the outer, too, which gives us minus two y squared. And then I for the inner too, which is minus X y and then l for the last two, which gives us why times y squared. 170. Algebraic Expressions Division: division of Algebraic Expressions. When dividing algebraic expressions, the falling formula is useful. This is often described informally as breaking up the fraction this formula generalizes to any number of terms. As shown in these examples here, when there is more than a single variable in the denominator such as these examples, we usually factor the expression and then cancel instead of using above formula. 171. Algebraic Expressions Example 2: start by factoring the new Mitter. Now the only factors of one or one and one. It's only possible factory ization of this problem is a one and a one. And since the sign Petters negative first and in positive Second, both have to be negative. And now cancel the factor X minus one on the top in the bottom, and we're left with X minus one. Instead answers d. 172. Algebraic Expressions Problem 1: multiplying the expression by using the foil method, we get X cubed minus next to the fifth plus two X and then minus to execute and in like terms. We get negative extra fifth on a positive one. Minus two is a negative one X cube plus two x, which is answer choice be 173. Algebraic Expressions Problem 2: working from the innermost parentheses we get five miles to is three plus. Why canceling the X is we now have now squaring the three. We get nine and nine times two is 18 distributed in negative. We get three minus three minus y and adding up like terms. These cancel gives us zero. So we have negative wide negative. Seven plus 18 is positive. 11. No, to stripping the Negev to We get positive too wide and negative 11 which is answer choice e . 174. Algebraic Expressions Problem 3: for statement one. We're told that a at B is equal to a B minus one. Now we know that ordinary multiplication is communicative. That what that is, we can change the order so that could be written as B A minus one. But this is precisely be at a so statement one is true which eliminates choice. Be no ver statement to notice in the definition here, A's and B's position. So the top becomes a times a minus one and the bottom is just a Then let's compare this result to the expression on the right side. We have one at one. Since one is Anais position, we get a one and since one is a beast position, we get a one here and one times one is one and one minus one is zero. And clearly, for most values of a, this expression will not equal zero. In fact, it will equal zero only when a is equal to one. Because then we have one times one minus one over one, which is one minus 1/1 or 0/1, which is in fact zero. But for any other value of A, it will not be zero, for example, of a is too. Then we have two times two minus one over two, which is four minus one over to or three halves, which does not equal zero now counseling the left hand side of statement three. We knows that he's this whole expression here is an A's position. So it's gonna pop out here and we get a at B times the next expression, which is C minus one. And now just apply the definition again to this expression inside the parentheses, which gives us a B minus one time see minus one, which is Distribute the sea, and we get a B C minus C minus one. Now for the right side of the expressionless work from the Interpret theses out. Since B is an ace position, we get be and C isn't B's position, so we get C minus one now going back to the Formula A is Anais position, but the whole expression BC minus one is in beast positions. So we get a Times B C minus one minus the one that's in the formula and distribute the A. We get a B C minus a minus one. Now, two parts of this expression are the same. The ABC is the same and the negative one is the same. But clearly the expressions do not equal each other because, see is not always gonna equal a. In fact, it will rarely equal it. Hence the answer is a 175. Algebraic Expressions Problem 4: Since these expressions are perfect square, try no meals. We get X Square plus twice the product, plus the last term squared. And for this when we get the first term square, which is four X squared minus twice the product, the product is eight. Dublin. It is 16 and then four squared is 16. Distributed negative sign on each term will change the signs and then add up like terms. X squared minus four X squared is negative. Three X squared and cancel u two's Here we have X plus 16 x is 17 axe and to add the 1/4 and the negative 16 and the negative 16 re