Transcripts
1. Intro: Hey, my name is Kevin Venter, and I would like to welcome you to this course on pre algebra, where we will be exploring the topics that will help you excel in pre algebra and help you build a foundation that will help you move on and master algebra. Before we start this course, I'm going to assume that you have a firm grasp on basic topics of math such as addition, subtraction, multiplication, division and fractions. If you need any help, refresh in your mind or some practice with those basic fundamentals of math, That's okay. I have you covered. You can take my course on quickly learning basic math where we cover each of those topics in depth. Before we start learning some of the amazing topics we will be diving into in a moment, I want to first see please eliminate from your mind idea that math is difficult. I know people start to get worried and intimidated when learning pre algebra and algebra, and that's my reason and motivation for creating these courses is to take all the concepts taught in math and simplify them. Anyway, anyone can learn I was someone just like you who struggled with math when I was in school, and after failing and more testing I can count. I developed a curiosity and hunger to really develop my math skills. And one of the things that I quickly discovered that no math teacher ever taught me is that math is not hard. It's simply like anything else we learning life, which requires us to learn certain rules and steps and practice once I was able to simplify . Or better yet, look at every math subject. From that point of view, I was able to pick up every math topic taught in grade school and college with much more ease, especially after putting in the extra work just how you're doing. By taking this course, I'm not going to bore you to death by give you endless lectures and this course, as in all my other math courses, you're going to learn with your senses by seeing and doing so. I encourage you to write the problems as I write them, and if you missed something I said or did, you can simply rewind the video and, after watching the video, do a few math practice problems that I will include at the end of each section, and I can assure you you'll be amazed at how you will learn math like you've never learned it before, so let's get started.
2. Real Numbers: I know I said this course or any of my other courses will not be endless lectures. But the way I like to teach is by first explaining the concept which will help you see the overall picture than doing several example problems to hope you fully understand, which will cement your learning. The first and most important overall topic in this course Israel numbers. What are real numbers? Aerial number is any number that can be located on a number line. What is a number line? Let's quickly draw a number line and place a zero in the center. On the right of the number line we have are positive numbers, like so one to three four five. The arrow indicates that the positive numbers continues towards the right, and on the left, we have our negative numbers, like so negative one negative, too negative. Three negative four negative five and so on. And if you have never dealt with negative numbers, they basically mean less than zero. And you may ask yourself, How is it possible to have anything less than zero? The way I like to look at it is if you have $100 in the bank. You buy an item online like let's see a gift card for one a one. When you look back at your bank account after you purchase of the gift card, it will say negative one, which means less than zero, and you need to place that amount in your bank account in order to bring the balance back to zero. So if you deposit $1 in your bank account, your battle it's will be zero. And if we bring back our number line, we can visually see that if we count from negative 120 it's one space forward towards the positive numbers. Great. At this point, you're probably wondering, isn't every number that exists a real number. But when we get into variables, you'll see why referring to numbers we find on a number line aerial number. So what are more examples of numbers we confined on a number Line six, which would be the continuation of our positive numbers if we wrote down more positive numbers on our number. Line 1.5, which is the same as 1.5 and is between one and two. Another would be negative ton if wrote down more negative numbers on our number line. Another branch within the realm of real numbers or type of number is rational numbers. What is irrational number? Irrational number is a number that can be written as a fraction and almost on numbers can be written as a fraction. By definition, we know that rational means reasonable, logical or even normal. And when we think about it, almost on numbers are rational and can be written as a fraction and found on a number line such as one to three 45 including negative numbers. Another type of number is irrational numbers. What is an irrational number now? Listing for a second, if rational, means logical, normal and reasonable. Irrational is the complete opposite. So we'll be an example of irrational numbers, a number that can't be rain as a fraction, such as pie, which you might have heard off and is represented by this symbol. Pi starts with 3.14 and the numbers never repeat themselves after the decimal point and goes on forever and can be written as a fraction. There's a lot of irrational numbers, but I don't want to get into too much detail because we really use them. I mentioned pot because it's pretty popular in the world of math and science, and I'm sure you're encountering. Sooner or later, another type of number is insiders, and this is a topic. We will be talking about a lot. And if you've never heard of the term integer, it basically means positive, negative zero or numbers with no decimal, which leads us to our next type of number, which is whole numbers. Whole numbers is basically positive numbers, such as 01234 56 etcetera. Next, we have natural numbers, which is basically the same as whole numbers, but we don't include the zero. Like if you're counting items in the real world, you don't start with zero. You just start with one to 34 etcetera. Another type of number, which you'll hear me mentioned quite a few times. And pre algebra and then algebra is prime numbers. Prime numbers is not something we were you specifically throughout pre algebra and algebra . But it's something that it's useful to know. So what is a prime number? A prime number is a whole number other than zero and one that is divisible by one and itself. I know that can sound confusing when you hear it or read it for the first time. But let me show you a few examples so you can see what I mean. The number two is a prime number. Why? Because it's divisible by the number one within the remainder, which would give us, too. You can also divided by itself because two divided by two is going to give us one. Another example would be three three, divided by one is three and three. Divided by three is going to give us one. Another example would be five five, divided by one is five and five. Divided by five is one. If you would like to see a long list of prime numbers, you go simply Koegel prime numbers, and you'll get an organized chart off prime numbers. You may ask yourself what is an example of a number that is not prime, and an example would be the number 10 and that's because it's divisible by two and five. I know we covered a variety of number types in this video, and I would like to draw a graph toe, help you map things out and put it into perspective. All the different types of numbers fall under the umbrella of Rhone numbers, and underneath the umbrella of real numbers, we have rational numbers under the branch of rational numbers. We have three different number types. We have fractions, decimals, which can also be converted into fractions, integers and, on the end of the spectrum, off row numbers. We also have irrational numbers like pi, which behaves strangely. I know we talked about a lot of different number types, but I can assure you that as we progress for this course, we will go deeper into number types, which will help you become better at pre algebra and then move on to algebra. That's going to conclude this video, and in our next video, we're going to learn all about the number line, which will help you fully understand the relationship between positive and negative numbers .
3. The Number Line: Hey, welcome back In our last video, we learned about real numbers. In this video, we will be learning all about the number line. We briefly talked about it in our last video, and if you've taken my basic math course we talked about it. We were learning about decimals. I know you're probably wondering, why would we be learning about the number line now if we talked about it when I presented it to you in the basic math course? And that's a great thing to ask yourself. One thing about the number line, which we really didn't get into much detail about before, is negative numbers. And we briefly touched on the topic in our last video when I was explaining the example of when you buy an item online for, let's say, $10 but only have $9 in your bank account. Your bank may still allow you to purchase the item, but when you look at your bank account after the purchase, you'll see you have a negative value of one. And that's basically what negative numbers are. It's a way of determining how far you are from zero and the beauty about a number line is that it's a great way to illustrate how far you are from zero cholesterol, our number line and placed 10 positive numbers on the right and to negative numbers on the left. Remember the error on the left and on the right of our number, line and the case. The numbers are infinite towards that direction. Another important fact to remember is that negative numbers are towards the left and positive numbers air towards the right. Now that we have our number line, let's try a few exercises you're bound to see on your test, such as plotting or placing a point on the number line. For example, you might get a question like place or plot, which basically means the same thing. So we'll use place since is more conversation like place a point on number seven? Immediately we see it's seven by itself, so it's a positive number. So we start at zero and we scan our number line. So we find the number seven and then we place a point. Great. Next. Let's police a point on the number three again. Let's start from zero as scan our numbers till we spot the number three and place a point. You don't always have to start from zero, but it's a good starting point, so we'll do it a few more times. Next, let's place a point on negative eight. This is the first time we're dealing with a negative number, and we can tell it's a negative number because they has a small horizontal line on the left . As you can see on the left side of our number line, where we have our negative numbers, each off are negative numbers. Has a small horizontal line on the left, same line we use when we subtract. So let's start at zero and scan are negative numbers till we spot negative eight and place a point. Next. Let's place a point on negative five. Just as before, Let's start at zero and scan towards the left so we see negative five and place a point far next and last exercise. I want to rewrite on number line so you can pause the video and redraw your number lines so you don't have so many numbers, and we can create some space between our numbers. Next, let's place a point on 2.5. If you're familiar with decimals and fractions. You know that 2.5 is the same as 2.5. So we place a point right between so 13 which makes it 2.5. That's going to conclude this video on the number line. If you have any questions, please don't hesitate to ask in our next video we're going to learn about greater than less than and equal to. I want to thank you for watching this video, and I look forward to seeing you in the next.
4. Greater Than, Less than & Equal To: Hey, everyone, welcome back. In our last video, we learned about the number line. In this video, we're going to learn about greater than less than and equal to. I want to start by saying this topic is not hard at all. I know I constantly say math is not hard, and it's simply the truth. It's all about knowing the rules in math and constantly practicing. The more you practice, the easier it becomes, just like speaking English. Think about it for a second. When you were a toddler, it took an immense amount of brainpower to create sentences and express yourself. But as you got older and spoke every day, talking became easier and easier. And it's the same with math because at the end of the day, that's all math is. It's a language. This is a topic which is based on common sense because it's pretty easy to spot when a number is bigger or smaller than another number. The only tricky part is using some of the symbols that are often used when comparing numbers. And the great thing is we will use a number line which we learned about in our previous video to help you visualize and see how bigger smaller in number is when comparing them. So what are the symbols that are used to compare numbers? We have the greater than symbol, which is like an arrow without the stick. We have the less than symbol, which is sort of like a cricket out when you think about it. And we all know the equal sign because it's always deals in math, no matter what level of problem solving your doing. So what would be an example of how these symbols are used in math? For example, let's look at these two numbers and try to determine which of the two is greater. Oh, we know three is greater, so we place a greater sign in between three and one. You might also be asked which of these two numbers is less than the other, and we know one is less than three. So we used the lesson symbol, which is like a bet or crooked l. But the number one would have to come first and the three after. Then we place our symbol because we read from left to right. I know this can be a bit confusing if It's your first time using these symbols to compare numbers, but I can assure you, once we go through a few more examples, you'll master this topic and the secret to knowing how to place. The symbol is the pointy part of the arrow is always pointed towards the smaller number, and the opening part of the arrow is always in front of the bigger number. Let me show you what I mean. Let's make the arrow bigger so you can see what I mean. As I mentioned before, the pointy part of the arrow is always pointing towards the smaller number, and the opening part of the arrow is always in front of the bigger number. I'm sure once we do a few more comparisons, you'll get the hang of this. Let's try another. Which of these two numbers is greater? 12 or nine? The answer is 12. So we place our greater than symbol like so. And as we mentioned before, the pointy part of the IRO points towards the smaller number, and the opening is in front of the bigger number. Next, let's look at which of the stool numbers is greater. We have 99 75. We know 99 is larger, so we place our greater than sign like so next. Let's switch things up a bit and look at which of these two numbers is less. We have 13 and 35. Which of these two numbers is smaller? 13. So we was are less than symbol like so. And as you can see, the pointy part of the arrow is in front of the smaller number, and the opening part of the arrow is in front of the bigger number. Next, let's look at 18 and 25 which is smaller. The answer is 18 so we use the less than symbol like so great. Next, let's switch things up a bit and look at 15 and 15. I know this is pure common sense, and we know that they're identical, so use are equal. Sign. Great. Next, let's bring back our number line, and you can pause the video and draw your number line so you can work along with me so we can explore comparing negative and positive numbers a bit further. What I would like to do is try a few more comparison problems. You wouldn't be asked on a test so far. First example, let's decide which is less negative. Five or two. The answer is negative. Five because it's negative, and if that confuses you, it's OK because at first glance you might say, Hey, five is bigger than two. But remember, as we learned in our last video, any positive number even one, for example, is bigger than any negative number, even negative 500. Let's try a few more, and I'm sure you'll start to get the hang of this. Next. Let's see which is greater negative one or negative three. The answer is negative one, because it's closer to zero. And as I mentioned before, the further a negative number is from zero. The smaller it is so we use are greater than sign in this case, where the opening end of the arrow is in front of the bigger number and the small pointy part is in front of the smaller number. Great far last example. Let's take a look at which is less negative. Four or four. The answer is negative. Four. Because its a negative number great. I hope that you've been able to learn and understand greater than less than and equal to. If you can't understand and aren't too sure of what was going on in this video, I urge you to re watch the video because repetition can always give you a deeper understanding of any given topic. I want to thank you for watching this video, and in our next video, we're going to really start to get into the more exciting parts of pre algebra by learning how to add integers.
5. Adding Integers: Hey, everyone, welcome back. In our last video, we learned about greater than less than and equal to. In this video, we will be learning a very important topic, adding integers. Remember in one of our earlier videos, when we were talking about real numbers, we said integers basically are positive numbers, negative numbers and zero, not including decimal numbers. In this video, we will be learning how to add integers, and I know you might ask yourself, I know how toe add very well. Why would I need to know how to add integers? And that's an excellent question to ask yourself because I asked myself the same thing when I learned how to add integers and is the primary reason why I said learning how to add integers is very important because negative numbers are involved and when adding negative and positive numbers, things can get a little tricky. But I can assure you that with the monitor problems we will solve in this video and a few tips and tricks I've learned along the way, you will be able to master adding integers and will be able to move on to the next topics in pre algebra and algebra. So let's get started before we start adding integers and learning the tips and tricks that will help us add negative and positive numbers. There is one thing we need to understand first, and that's absolute value. What is absolute value? Absolute value basically is. And let me give you a few examples with the numbers. Since this is a math course as opposed to giving you a working explanation, let's start with the absolute value of number seven. If you've ever seen these two bars on need side of the number, it's a symbol that just tells you you're dealing with absolute value. So don't let it intimidate you. So was the absolute value of number seven. It's seven. Another example would be the absolute value. Off Number two is simply too another would be. The absolute value of five is five, as we just saw before. Now let's look at a negative number since I know negative numbers as they have no concept to many of you taking this course, the absolute value of negative three is positive. Three. I know that can be a bit confused, but I can assure you in a second I will visually show you on a number line. Why a negative number translates of the whole positive number. It represents one dealing with absolute value. So let's look at one more. The absolute value of a negative six, as you might have guessed, is positive. Six. So now you know, when dealing with absolute value, no matter if the number is negative, it's going to take away the negative symbol and translate toy positive number. Now let's bring back our number lines so you can see why this occurs with absolute value. You can pause the video and draw your number line so that you can work alongside with me Now that we have our number line. Let's start with absolute value off number seven, as we did before. So as we said, the absolute value off seven is simply seven. And what this is really telling you is the distance from zero. And that's why when we're dealing with a negative absolute value, it six away the sign and simply tells you what is your distance from zero. So let's keep going so you can see what I mean. The absolute value off to is simply, too, because if we count from zero to the number two. You'll see there are two places from zero. If we look at the absolute value of number five, we learned it's five. And if we look at our number line, it's five places from zero. Next, let's look at negative three. And this is where you see why the negative Sim was taken away when trying to determine the absolute value of the number because absolute value is determining the distance of the number 20 whether it's positive or negative. And if we come from negative 3 to 0, we simply get three. Next, we have negative six, and if we look at it on our number line, we can see it's six places from zero. So that gives us six. I know at this point you're probably wondering why is it important to know absolute value? And the reason is we will be adding and subtracting negative and positive numbers, and it's important to look at a number for its whole value. Temporarily take away the symbol, and after you add or subtract, determined where to put the symbol, and that's very important in algebra. So no need to worry we're going to solve enough problems where you'll fully understand how toe addis attract integers. So let's get started adding integers. But first, I would like to start with some basic guidelines or basic rules, if you will, to adding integers. Let's clear our screen so that we can go over the basic rules of adding integers. So what are the rules of adding integers? The first rule I would like to go over is adding a positive number toy positive number. And I want to put my positive and negative symbols and parentheses just so we won't confuse them with the text. So what happens when we add a positive number with a positive number? We get a positive number and just so we don't get confused, the hashtag is dissemble for number and I ask it a bear with me because in a moment once we get past these basic rules, we could start applying them and seeing why they are important. So let's do a bit of arithmetic with the rule that we just learned. First we have four, which is a positive number plus too, which is another positive number is going to give us six which is also positive because a positive plus a positive gives us another positive number. Next, let's look at seven plus three, which is going to give us 10. And it's positive because a positive plus a positive gives us a positive. As the rule states. I know you're probably wondering this is extremely basic and most likely you already know this. But you'll see in just a second while breaking it down like this because when we add negative and positive numbers, things can get a bit tricky for next rule we have. If we add a negative number toe, another negative number is always going to give us a negative number. And as we learned before, we were learning about negative numbers and looking at examples on her number line, we learned if we have negative three and we add negative three to it, that would give us negative six. I know that can be a bit confusing, so let me give you a nice visual example with pencils. Let's say one day you go to school and you have a test, but you forgot your pence went home. So you asked your classmate to lend you a pencil, so he lends you one pencil. Let's say after your test you forgot to give your classmate that pencil. Now you all your classmate. One pencil since you didn't give it back. So in a way, negative numbers is the same as going. Now let's say on the next day you come to class and you forgot your pencil again and you ask your same class me for another pencil. And since your classmate is a nice person, he lends it to you. But you again forget to give it to your classmate after class. Now you know your classmate two pencils, because negative one plus negative one equals negative to. And as the rule states, adding to negative numbers is going to give us a negative number. Let's look at a few more examples because that's how this will become easy by repetition and practice. Let's add negative four plus negative three, which is going to give us negative seven. Next, Let's look at negative six plus negative, too, which is going to give us negative eight because, as we just learned, when we add to negative numbers, you get a negative result as your answer and just to show you why this occurs. Let me show you on the number line. Let's take away a few positive numbers and add more space on the left where we have our negative numbers. So we're starting at negative six, and we are adding negative, too. One to gives us negative eight. Great. I hope this gives you a little more clarity on adding to negative numbers for next rule we have. If we're adding a positive toy negative or a negative toy positive. The good thing is you do the exact same thing in both cases. Onley tricky part is you have to figure out whether your answer is a positive or a negative after you add. But the great thing is that it's very easy to figure out once you've done it a few times, and the trick is, your answer will be the same sign or symbol as the larger absolute value and your problem. I know that can be a bit confusing, but I can assure you, after we solve a few problems, you'll understand what I mean. So let's look at a few examples, and I want to use basic numbers so you don't get caught up in the calculations but rather just learned the rules and everything else becomes easier. For first example, let's look at three plus negative two that's going to give us one. Now we know it's a positive one because, as we just learned that, we always take the sign of the bigger number, which in this case, is positive. Three. Next, let's look at six plus negative four. That's going to give us positive two. And as you can start to see when we're adding a positive and a negative, you always subtract and simply take the sign from the bigger number, which in this case is six. And we have positive, too. And I know you're probably wondering. We aren't adding a sign or taking it from the bigger number, but we only placed a sign when it's a negative number. When a number has no sign around it, you know it's positive. Next, let's look at negative four plus seven. And as we just learned when we're adding a positive and a negative, we simply subtract and then take the sign of the bigger number. So negative four plus seven is going to give us three and seven is the bigger absolute value, so our final answer is positive. Three. Great. Let's switch things up a bit and look at what happens when we add a bigger negative number to a smaller positive number. Far next problem. We have negative nine plus two. The same rules apply. We still subtract. So negative nine plus two is going to give a seven. And as we learn before we take the sign of the bigger absolute value, which in this case is nine and we have a final Valley of Negative seven green. For less example, let's take a look at negative eight plus five that's going to give us three. We see that eight is the bigger absolute value. So we bring this symbol over, and that gives us a final answer of negative three Greek. I hope you're starting to see and get the hang of adding integers. It can be a bit tricky at first, but I can assure you the morning practice, the easier it gets in our next video, we will be subtracting integers. I want to thank you for watching this video, and I look forward to seeing you in the next
6. Subtracting Integers: Hey, everyone, welcome back. In our last video, we learned how to add integers in this video. We're going to continue with what we learned in our last video and learn how to subtract integers. The great thing is, we've already been subtracting integers, which we first got a chance to do in our last video. There just a few more tips and tricks you need to learn with subtracting integers and there won't be a problem out there regarding into jurors. Worry you won't be able to solve it after you turned them. And just to give you an example of how we've been subtracting integers, let's take a look at seven plus negative five. And as I said before, when you have a positive and a negative, you simply just subtract and then take the sign of the bigger absolute value. So this is going to give us positive, too. I quickly want to note that this would also be the same as trying to solve seven minus five , which we know is, too, and that's what we would normally see. But as we get into other topics in pre algebra and algebra and integrate variables, you start to see parentheses and letters which hold values, and you'll start to see the true power of algebra and really start to master math. But for now, let's continue with subtracting integers. Let's look at another case where we have a positive minus another positive. We know that's going to equal a positive. Let's look at a few examples, such as six minus two. We know that's going to give us four. Another example would be a minus three, which would give us five. Another example would be 10. Minus nine is going to give us one. And this is super easy stuff, which you've been seeing since the first grade. And I'm doing these very basic examples to point out the fact that the first numbers are larger and the second numbers or the number which we are subtracting from the bigger number is smaller. And that's how we've always learned how to subtract. However, when you get into algebra, that changes because we're now dealing with positive and negative numbers, and we saw an example of that in our lads video. When we were adding integers. I made the example with the pencils, but I would like to show you a few more just to refresh your memory and so you can see what happens when you try to subtract a bigger number from a smaller number. For first example, let's take a look at four minus five and quickly you notice that the first number is smaller and the second number is bigger. So what is four minus five that's going to give us negative one? And to show you why that occurs? Let's bring back our number life so you can visually see what happens when you try to subtract a bigger number from a smaller number. We're starting at four, and we're taking away five. So we count five toward zero because we're subtracting. If we were adding, we would go the opposite way. So let's count one to three four five and you see we arrive at negative one. I know this can be confusing when you see it for the first time, especially if you're not too familiar with the negative numbers and it's OK. Let's go over a few more examples so you can get the hang of this. But I quickly want to point out before we move on to our next example, which we saw a lot of examples when we were adding integers is four minus five is the same as this. And remember, I said, When you have a positive any negative, you always subtract. And after you subtract, you take the sign of the bigger absolute value, which in this case is five. So we get negative one. After we subtract. I hope you were able to make that connection. Let's take a look at another example far next problem. Let's take a look at two minus seven. We learned that when the first numbers smaller, we get a negative result, so this is going to give us negative five, and a good way to test this is to grab my calculator and subtract a smaller number from a bigger number and vice versa. A bigger number from a smaller number and you'll see in the case where you subtract the smaller number from the bigger number, you get a negative result, and when you subtract a bigger from a smaller number, you got a positive. As a result, the best way to see what's happening is with a number line, so let's quickly see how we arrive at negative five was starting at Sue because two comes first, which means we're taking seven from two, and then we count towards the negative numbers. Since we're subtracting one to three, four, five, six, seven and we get negative five Green and I want to quickly point out that this problem is the same as this, which you might see in your test or in a math textbook. Let's look at one more example like this because we know that practice makes perfect, especially when it comes to math. Let's take a look at one minus three that's going to give us negative, too. And let's quickly bring back our number line and see how we arrive at negative two. We're starting at one and taking away three one to three, and we arrived at our answer. Negative, too. And as you might have guessed, one minus three is the same as this, which you might normally see on your test or in a math textbook. Now let's look at another case, which is where I really, really want you to pay close attention because things can get a bit confusing here, and it's the case when you're adding to negative numbers. You add them, but they remain negative. For example, negative two minus three, which we've also seen. Looks like this. And let's bring back our number line so you can see what occurs when you add them. But they remain negative were starting at negative two and adding negative three. One to three, and we arrive at negative five. Next, let's look at negative one minus six, which is going to give us negative seven. Remember, we're starting at negative one and counting six one to three, four 56 Great. I wanted to quickly refresh your memory with some of what we saw in our last video, because this is where things can become a bit confusing. And this one we're subtracting a negative number as a post, adding them, and you might tell yourself, is this looks exactly like what we did before, but no, there's a slight difference here. There's an extra minus sign, but we use a parentheses. But the fact is, there's another subtraction or negative symbol between the numbers. So what happens in this case? Let's look at seven minus negative. Three. Both subtraction or negative science cancel each other out and become positive and just to clean it up. Let's re write the problem and it becomes seven plus three, which is going to give us 10. I know that can be a bit confusing. I know the first time I learned this, it was very frustrating because it just didn't make sense. But through numerous examples, I was able to understand it. So let's look at a few more examples for next problem. Let's look at five minus negative, too, as we learned before, when we're subtracting and negative number, the science cancel each other out and it becomes addition. So five minus negative two becomes five plus two, which is going to give us seven green. Let's look at another example for next problem. Let's take a look at nine minus negative four. We learned in a case such as this one, when we have to negative signs and a negative number separated by parentheses, they cancel each other out and become positive, and the parentheses doesn't really hold any value but its place. So the two negative symbols don't get mixed up and look like one. And when we get to order of operations, you'll see the real purpose of parentheses. So we were right. This problem and it becomes nine plus four, which is going to give us 13 great far. Next problem. Let's look at what happens when we have negative three minus negative four. First thing we do is castle or cover are two minus signs, which is going to give us negative three plus four. And remember, I said earlier, when you have a negative any positive, you always subtract and take the sign off the bigger, absolute value, which in this case is four. And this is going to give us positive one. Great. Let's look at one last problem similar to this one before we conclude this video. Far last problem. Less sticky. Look at 10 minus negative six. And the first thing we do is cancel out or convert our subtraction symbols into positive symbols. And then we were right. The problem just to clean things up a bit and we're left with negative 10 plus six, which is going to give us negative four, because when we have a positive and a negative, we subtract and take the symbol of the bigger absolute value, which in this case is 10 great, and that's going to conclude this video on subtracting integers. I know subtracting integers can be a bit confusing, but I encourage you to watch this video two or three times and do as many practice problems as you can, and I can assure you you'll get the hang of this for next video. We will be learning how to multiply integers. I want to thank you for watching and look forward to seeing you in our next video. Thanks.
7. Multiplying Integers: Hey, everyone, welcome back and our last few videos we've learned about Roe numbers, the number line greater than less than and equal to adding integers and subtracting integers. We're making great progress so far, and you should feel great about that in this video. And in our next video, we will be learning how to multiply and divide integers, which I could say from experience is easier than adding and subtracting into jurors. And I know that might not make sense at first, because when you're adding and subtracting basic numbers, multiplying and dividing is usually more complex. But in the case of pre algebra algebra, it can be a bit easier. And let's look at why we just have to learn. If your rules first and then you'll definitely be able to master multiplying integers. After all the problems and examples we will be solving together, the first thing I want to point now is the fact that there are a few ways to write a multiplication problem, and your first thought might be why not just write it one way and be done with it? But as I mentioned before, we're getting into order of operation. We will start to add, subtract and multiply all in one problem. And when you're doing several calculations in one problem, it's more efficient to write your multiplication in sort of a shorter version. So it takes up less space and causes less confusion. So what are the different ways to write a multiplication problem? One way, And this is the format you're probably most familiar with is with an ex between the numbers , like so five times five is going to give us 25. And this method is not really used in pre algebra algebra because, as I mentioned before, when we start to use variables, which are letters that hold values, and X, which is a letter can confuse you because you might think it's a variable. When are you trying to do is multiply another way you can write a multiplication is what they got like so three times seven, which is going to give us 21 and you can already see that the dot looks more efficient or cleaner than the X another way. And the method that is used in pre algebra and algebra is with a parentheses, like so here we have two times six. And I know this can be a bit strange at first, because you have two numbers that are just next to each other in the parentheses. But I can assure you, once we get there are enough examples, this will seem perfectly normal. Just takes a bit of getting used to. And, of course, two times six is going to give us 12 before we start multiplying. We have to go three funerals as we did when we were adding and subtracting integers because we're now dealing with positive and negative numbers. So let's look at our first rule, which is one. You multiply a positive number by another positive number that's going to give us a result of another positive number. Let's look at a few examples. For example, one times five is going to give us five. Another example would be four times two is going to give us eight. I also want to quickly point out the fact that if one number is inside the parentheses and another number is not, you still multiply because in algebra, it's assumed that when two numbers are next to each other without a symbol, you multiply. For example, six times seven, which is going to give us 42 green. Let's now look at our second rule, which is when you're multiplying a positive number and a negative number, or vice versa. A negative number. Any positive number. Your result will be a negative number. For example, let's take a look at six times negative, too. That's going to give us negative 12 because when we're multiplying a positive, any negative are. Result will be a negative number. Let's look at another example negative three times five, Which is going to give us negative 15. Great. Let's look at another example before we move on to our third rule. Let's look at nine times negative seven. This is going to give us negative 63 as a result. Great. Now let's look at our third rule, and it's when you're multiplying to negative numbers. The result will be a positive number. Let's look at a few examples. For first example, we have negative seven times negative three, which we just learned what we multiply. Tonegative numbers. We get a positive result, so this is going to give us 21. Let's look at another example negative eight times negative. Four. This is going to give us 32 Great. Let's look at one more example before we get into a few practice problems, which will help Postmaster multiplying integers for next problem. Let's take a look at negative five times negative seven. This is going to give us positive. 35. Great. Now let's take a look at a few practice problem just so we can reinforce multiplying Integers for first problem. Let's take a look at six times five, which is going to give us 30. Next, Let's take a look at seven times three. And remember, we said earlier, Even if one number is outside of the parentheses, we still multiply. And before we end this video, I will give you an example of why this occurs by briefly looking at variables and exploring what are variables. So seven times three is going to give us 21 Great for next problem. Let's take a look at negative four times 10 which is going to give us negative 40 for next problem. Let's take a look at negative two times negative 12 which is going to give us positive 24 because when we multiply too negative numbers. We get a positive result far. Next problem. I want to add another number. So instead of multiplying two numbers, we will be multiplying three numbers, and I don't want you to let the extra number intimidate you. You'll see this on your test and in your algebra textbook, but it's just that simple. We just need to go step by step from left to right, like we read words off a paper. So we start by multiplying two times one which is going to give us, too. We cross out the 211 so we know that we multiply those two numbers. Then we proceed to bring down our three next tour to and we multiply. And for a last step, we multiply two times three, which is going to give us six. And that's our final answer. I know when you first see something for the first time, it can be challenging to grasp it. So let's try similar problem. Let's multiply seven times zero times nine. And as we did before we multiply from left to right. So seven times zero is going to give us zero. We bring down our zero and bring down or nine, and we multiply zero times nine, and that's going to give us a final answer of zero for next problem. Let's take a look at two times negative five times negative eight and just says before we multiply two times negative Phi first, which is going to give us negative 10. Next, we bring down our negative eight and multiply negative 10 times negative eight, which is going to give us 80. I hope you were able to understand how to multiply. Integers with the various problems will work through, and I always recommend it's a good idea to re wash the video a few times if there was something you didn't understand. And after you watch the video a few times, it's good to try. If you practice problems before we conclude this video, I want to briefly touch on the topic of variables because, after all, this is a pre algebra course, which is what prepares you for algebra and algebra. Variables are used a lot, so it's best we start conditioning our minds and get familiar with them little by little, so that by the time we get to algebra, you feel comfortable with variables. So what is a variable? A variable is when you simply use a letter to store a value. For example, here we have the letter eight, and we're going to store the value of five in the letter. A. So this basically means when you have a problem and you see the letter A or variable ate, it means five, and it can be any letter on the alphabet. I'm a programmer, and we use variables as well. When we were writing programs to store values and variables and that concept comes from algebra, let's look at another example of a variable here. We had the letter B, and we're going to be storing inside this variable negative, too. So any time we see be in our program, that's simply equals negative, too. Let's look at one more variable before we try a few practice problems, and we're going to use the letter C and store the Valley of 10. And I know you're probably wondering, why not just write the number instead of using a variable? And that's a great question, and there are several reasons why. But a primary reason is if you have the Valley of let's say 20 million and you have to multiply. That number with a few other numbers is busted. Stored in a variable. I use the variables instead of the value. Let's take a look at a few examples so you can see what I mean. Far first problem. Let's take a look at three times a. I don't remember. We learned earlier that in algebra, when two numbers are next to each other, you must apply them. And yes, the letter, which represent a variable, is a number. When you think about it, we have three times A. When we look at what is stored in the variable A. We know it's five, so we know this is the same mass three times five, which is going to give us 15. Great. Let's try another. Let's look at seven times be. We know B has a value of negative, too, so we can rewrite this as seven times negative, too, which is going to give us negative 14. Let's look at five times seat. We know C equals ton, so this can be a river in as five times 10 which is going to give us 50. I wanted to go to a few basic problems so you can see how variables are used. I just want to quickly due to more problems, and they form of a full expression, which is how you would normally see it on your math test or your math textbook. Let's look at four times C plus six. You don't need to always have to rewrite problem, but it helps if you define your variables when your first working with variables to help reduce confusion. So we can rewrite this problem as four times the value of the variable, which is 10 plus six. Now we can solve our problem, and we've done this numerous of times before, so we start with four times 10 is going to give us 40. We bring down the six and we add 40 plus six, which is going to give us a final answer. 46. Great. Let's try one more problem before we conclude this video. Far last problem. Let's take a look at two times a times be minus C. First, let's redefine our variables and give them their numerical values to avoid confusion. And we can rewrite this problem like this and now we can solve it two times five is going to give us a ton. We'll bring down our negative two and negative turn and we do 10 times Negative two, Which is going to give us negative 20? No, we're bring down our negative 10 and this is going to give us a final answer off. Negative. 30. Great. This is going to conclude this video on multiplying integers and our next video. We will be learning how to divide integers. I want to thank you for watching. And I look forward to seeing you in our next video. Thanks.
8. Dividing Integers: Hey, everyone, welcome back. In our last video, we learned about multiplying integers, and that was a video that was a bit longer than usual because we covered a lot of practice problems and introduced several new concepts like variables. In this video, we will be exploring Dividing integers, which is the fourth pillar of math, the other three being adding, subtracting and multiplying. So Dividing insiders is just as important in our last video, where we were multiplying, integers winner several rules regarding negative and positive numbers, and the great thing is, those same exact rules apply when dividing integers. And just like there are different ways you can write your multiplication. There are several different ways you can write your division as well. Let's take a look at a few examples. One way we're used to seeing a division problem is like so eight divided by four is going to give us, too. Let's look at another example of how you can write your division problem, and this is how you would normally divide when you're dividing on paper. But you rarely see a division problem given to you in this format and, of course, nine divided by three is going to give us three. Another way you can write your division problem is in the form of a fraction like so here we have 20 divided by fun, which is going to give us four. Now let's look at a scenario on what occurs when we're dividing and we're dealing with positive and negative numbers. Scenario Number one would be the same signs, for example, both the numerator and denominator or positive. This is going to give us a positive result, just like when we're most applying. Let's look at a few example problems. Let's look at 10 divided by five, which is going to give us too. Another example would be six divided by two. This is going to give us three. Another scenario. Where we have the same sign is If both numbers are negative, we still get a positive result. Let's look at a few examples. We have negative 15 divided by a negative three. This is going to give us positive five. Another example would be negative. Eight divided by negative, too, which is going to give us four. Now let's look a scenario to which is when we have different signs. Let's look at a few examples when the numerator is negative. Here we have negative six divided by one. This is going to give us negative six. Another example would be negative. 100 divided by 25. This is going to give us negative four. Now let's look at the same case. But now the numerator is positive and the denominator is negative and every doesn't make a difference. The result is still going to be the same, but I wanted to still show you what occurs. So let's look at a few examples. Let's look at 18 divided by negative nine, which is going to give us negative, too. Let's look at another example 25 divided by a negative five, which is going to give us negative five. Now that we know that different scenarios, we can encounter one. Dividing integers. Let's now go through if you practice problems so you can fully understand dividing integers for first problem. Let's take a look at nine Divided by three. This is going to give us three. Next. Let's take a look at 10 Divided by two. This is going to give us five. Next. Let's take a look at 24 divided by and negative six. This is going to give us negative four. Next. Let's take a look at 30. Divided by negative three. This is going to give us negative 10. Next. Let's take a look at negative 50. Divided by negative 10. This is going to give us five. Next. Let's take a look at negative 40. Divided by a negative five. This is going to give us eight. Next. Let's take a look at three times two, divided by two. First we multiply three times to which is going to give us six. Now we divide our six by two and we get a final answer off. Three. Next. Let's take a look at 10 times 40 divided by negative four. First we multiply 10 times 40 which is going to give us 400. Next we divide 400 by negative four, which is going to give us a final answer off. Negative 100. Next, let's take a look at four divided by a negative four times five. First we divide four and negative four, which is going to give us negative one. Next we multiply negative one by five, which is going to give us a final answer off. Negative five. Great. Next I want to do to problems with variables which is going to be great practice from We get its order off operation. Let's take a quick look at our variables. A is going to equal negative four b is going to equal to and C is going to equal six for first problem. Let's take a look at seven times a divided by beat and as we did before when we were multiplying integers and we first looked at, variables were right. Our problem for clarity and this problem can be rewritten like so. First we multiply seven and negative four, which is going to give us negative 28. Next we divide negative 28 2 which is going to give us negative for teen and far less problem. Let's take a look at 10 times, be divided by a and we can rewrite this problem like so. First we multiply 10 times to which is going to give us 20 and to get our final answer, we divide 20 by negative four, which is going to give us negative five great, and that's going to conclude this video on dividing integers. In our next video, we will be looking at power exponents, which is fairly easy. And if you've made it this far, I'm confident we could get by it together. I want to thank you for watching and look forward to seeing you in our next video. Thanks.
9. Powers & Expoents: Hey, everyone, welcome back. In our last video, we covered dividing integers. In this video, we will be exploring the topic of powers and exponents. I want to quickly note that these terms are interchangeable. Sometimes you'll hear exponents, and sometimes you'll hear power, but they both mean the same thing. So what is an exponents or a power? It's basically a shortcut to multiplication when you're multiplying numbers by itself the same way you learned that when you're multiplying, it's a shortcut for addition. So when you have a number and you need to multiply by itself is easier. If you write it as an exponents, let's see why. Let's say we have the number two and we want to raise number two as a power or given exponents off to when you haven't exponents of two. It's also called square, so this can also be referred to us. Two. Squared A quick note before we saw this exponents, and you might be asked this on your tests, or it could come up as a question. During your math class, this bottom or main number is called the base, and the top number is called the Exponents, or power and I give you these definitions because you can be asked, what is the exponents of this problem or what is the base, or what occurs when two numbers have the same base? So what does this mean when you have two squared or to raise to the power of to What does basically is telling you is two times two, which we know is four, and you can just write two times two when you're making a calculation. But remember, we said we use exponents as a short cut. So two squared is a much cleaner way to express two times two and there a few more reasons why it's better to use exponents and powers, and in a few seconds you'll see why. And just so we're clear. As we solve more problems like this together, I want you to know that the base in this case, Number two is the number one most applied by itself, and the power is how many times we multiply the base by itself. So let's look at another example for next example. Let's take a look at four. Race to the power of three. I remember we said the base, which in this case is four is the number we most applied by itself. In other words, the number that repeats itself and the power is how many times repeated. And since the powers three, we repeat 43 times and this becomes four times four, which is 16 times four, which is going to give us 64. Let's look at another example, three to the fourth power We know the power is how many times we multiply the base, which is three. So this becomes three times three times three times three, which gives us 81. And this is when you could start to see how using exponents can be a shortcut to multiplication. Let's look at another example, but in this example, we're going to multiply two numbers with exponents and a quick rule when multiplying two numbers that have exponents and the base is the same. We simply add the exponents and keep the base. Let's see what that looks like. Let's say we have two squared times to tow the fourth power, and as we just learned, the rule states. When the base is the same, we simply add the exponents and keep the base and as we can see, the base is the same. So we have to do is add the exponents, which is two plus four, which is going to give us six. We kept the two, which is the base, and we have our final answer and you don't have to write these steps like this. You can just keep the base and at the exponents in your head. But I wanted to visually show you what occurs, and this is called simplifying in math. If you took my basic math course or remember when you were learning fractions in your math class, you learned simplification. You usually simplify your numbers or problems in order to make it easier to solve. Because if we think about it, two squared and two to the fourth Power is the same as two times two, and to to the fourth, power is to multiply by itself for times which looks like this. And since the base is the same, this would be like just bringing all these numbers together and just adding a multiplication sign and multiplying as if the exponents was six, which would give us a final answer of 64. I know this can be a bit confusing if this is your first time seeing this. So let's try another. For next example. Let's take a look at four to the third power and four to the fourth power. And as we learned before, when the base is the same, we keep it and add the exponents. So we move over four and at three plus four, which is going to give us a final answer of four to the seventh power. Great. Let's look at another example. One. We have two different bases. For next problem. We have three squared times, four to the fourth power. In this case, the basis are different, so this can be entered by just writing simplified. And as we started to simplify mawr exponents in a second, you'll see why you can just write simplified when the basis are different. But first, let's go over another important rule when dealing with exponents. And the rule is when you're dividing numbers with exponents that have the same base you subtract, which is different from adding them when we're multiplying exponents, and this is easy to remember. If you're multiplying exponents and the base is the same. You add the exponents. If you are dividing exponents with the same base, you subtract exponents. Let's take a look at what that looks like. And I don't want you to let this intimidate you. It's pretty simple and straightforward. Let's say we have three to the fifth power divided by three squared, and when we have a problem like this, we were top to bottom and we realize we have the same base. So we keep the three as a track of the exponents. As the rule states, when you have the same base and you're dividing, keep the base and subtract exponents. So five minus two is going to give us three. And our final answer is three cute and let's break this down so you can visually see why this occurs which will give you a deeper understanding. So let's look at three to the fifth power over three squared fully expressed. I want to quickly remind you. Remember, we said exponents is a shortcut for multiplication. So even though we're dividing, our exponents are still multiplication within themselves. I wanted to fully express our exponents in this fractional format so you can see why we subtract our exponents when we're dividing toe exponents where the base is the same, and as we can see, all these numbers are multiplied on the top and multiplied on the bottom. But also remember, when you're dividing toe identical numbers, the result is one so three divided by three is one, so it cancels itself out. Next, we have three divided by three again, and that cancels itself out, and we are left with three threes, which represents our final answer. Three to the third power. I know this can be a bit confusing, especially if you don't remember simplify fractions. When you're dealing with fractions and the numerator and denominator are the same, you can dry line over them, and that sort of eliminates how we did with our threes. As we get deeper into algebra, you'll start to see more and more why that occurs. Let's look at another similar example so you can get the hang of this. Let's take a look at five to the seventh power divided by five to the fifth Power, and we learned, as the rule states, that when the base is the same, all you have to do is subtract their exponents when dividing powers. So we keep five as the base and calculate seven minus five, which will give us a final answer of five squared. Let's take a look at one more example. Before we get into our practice problems for next problem. Let's take a look at a squared divided by a squared. And since we're dividing, we know we keep our base as subtract our exponents. So this becomes eight with a power of zero. And here we have a very simple rule. When you haven't exponents of zero, that's just going to equal one great before we conclude this video, I would like to do if you practice problems just so we can bring together all the things that we have learned with powers and exponents. Far first set of problems, and this is a question you're likely to see on your test. Fully express our exponents for first problem. Let's take a look at 10 to the fourth power. When fully express, this simply gives us 10 multiplied by itself four times like so next. Let's take a look at too cute, which looks like so great, pretty easy stuff. Want to get the hang of it. Next. Let's write our fully expressed powers as an exponents. Let's take a look at three times. Three times three. What would this look like as an exponents? We know three is the base and amount of times you multiply. The base is the exponents. So this is three to the third power. Far. Next problem. Let's take a look at seven times. Seven times, seven times, seven times seven. We keep our seven and count them out. We multiply it and place that as the exponents for an answer of seven to the fifth Power. Great. This is going to conclude this video on power and exponents for next video. We will be learning about order of operation, which I consider one of the most important topics in algebra. I want to thank you for watching and I look forward to seeing you in our next video. Thanks
10. Order of Operation: Hey, everyone, welcome back. In our last video, we covered powers at exponents, which is pretty easy stuff. What? You get the hang of it. In this video, we will be covering one of the most important topics of pre algebra and algebra. Order of operations. I want to start by saying, Don't let the title off order of operations scare or intimidate you. If you know how to add, subtract, multiply and divide. This will be very easy. Order of operations is simply a collection of addition subtraction, multiplication division exponents which we learned in our last video and variables which you encounter every now and then. All you need to know which is the case with any math problems or topic is what are the steps in rules to solve a particular type of problem and enough practice. First, let's start off with the most important rules when solving order of operation expressions, and I want to emphasize that when solving a problem in the realm of order of operation, you need to follow these rules and order. I recommend you memorized them because they will help you save a ton of time when solving these type of problems, and I'll show you a little trick, which I still use till this day when solving or of operation problems called penned ass, which you've probably heard of before. But first is important to go over the rules. So rule number one solve everything inside the parentheses and brackets first. I know that can sound a bit confusing efforts, so let's look at a few examples so you can see what I mean. For first example, we have two times a pair of brackets, which contains five plus two minus three. I remember our first rule is solved everything inside the parentheses and the brackets. So in a case like this, you may ask yourself which, though I saw First and the answer is, you always want to solve the innermost parentheses or prayer of brackets, where we have five plus two and a quick note. Parentheses and brackets are the same thing or have the same meaning in order of operation . The reason why we use both is to help distinguish and separate the numbers, because if you have to parentheses in front of each other, you might get confused as opposed to if you have parentheses and then a pair of brackets or even curly brackets. And as we solve more problems together, you'll see what I mean. So let's solve this problem. And as I just mentioned, we solve the innermost parentheses or pair of brackets first. So five plus two is going to give us seven. Next, we rewrite our problem with the numbers that are left in the same form a in which we see them. So this now becomes two times seven minus three. And as we learned, we solve what's inside the parentheses first. So we do two times seven, which is going to give us 14. And since we don't have any more numbers left in our parentheses, weaken. Just write it by itself and bring down our minus three. And we subtract three from 14 to arrive at our final answer, which is 11. So, as you can see, part of operation can look a bit intimidating at first. Once you know the steps, it's fairly easy. Just takes a little practice, and that's why we're going to do several problems so you can get the hang of this next. Let's look at our second rule, which states express all exponents or do our exponents. For example, let's say we have five plus three squared. Your first instinct might be to add five plus three, which will give you a and then say you have a square. But that's wrong. As a rule, states fully express your exponents first. As we learned in our last video, a number that is squared simply means you multiplied by itself. So three squared becomes nine because three times three is nine, and now we have five plus nine, which is going to give us 14 Great. And if you have a scientific calculator, you can try this all for yourself. Easily process. We can use a calculator in our math class or during our test is crucial that we know these steps. Now let's move on to our next rule. Solve your multiplication and division from left to right. For example, let's say we have 10 plus five times four, and as the rule states, we first soft the multiplication and division first before we saw the addition or subtraction portion. So first we do five times for which is going to give us 20. Next we bring down our 10 and add 10 plus 20 which is going to give us a final answer off. 30. As you can start to see sort of operation isn't difficult at all. It's just a matter of learning the simple rules and doing them step by step. Let's look at our fourth rule, which states solve your addition and subtraction from left to right. Let's look at an example. Let's say we have five plus 20 minus 10 and as we just learned, we do the addition first. So we add five plus 20 which is going to give us 25. Next, we simply subtract 10 from 25 which is going to give us a final answer of 15. And there you have it, the four rules that govern order of operations. And I want to reiterate what those rules are and show you how you can easily remember them by simply remembering the word penned. As so, Let's reiterate so far, first step was scanning. An order operation problem is to solve everything inside the parentheses and brackets, and it doesn't matter if you have a multiplication addition or subtraction. Anything that resides inside the parentheses always get solved. First, Second express all exponents. In other words, make sure you solve or know the true value of your exponents. Next, Arthur Step is solved. Any multiplication or division in your problem, from left to right. And last but not least, we saw any addition or subtraction in our problems. So as you can see a lot of operations is four simple steps. And to make it easier, I want to show you a little trick I learned a long time ago and still use until this day. And that's Pandas, which is an acronym for these four rules, which means parentheses, exponents, multiplication division addition and subtraction. I would like to do as a reminder, because it may be hard to remember these rules when you're starting out with order of operations. I write this work on the corner of my paper and quickly scanned through each letter in the word when I'm solving order of operations problems, as we will do as we solve a few practice problems so you can get the hang of this and you don't have to do this. But it's just a little reminder of what the rules are to help you solve problems consistently so Let's take a look at our first problem and we will start with a few easy ones and little by little Adam or anymore to our problems, which will serve you as good practice. Let's start with 11 minus three plus six, and this is a fairly easy order of operation problem, and we will quickly scan AARP MGA's indicator and check. Do we have parentheses? No exponents, no multiplication, no division, no addition. Yes, and subtraction. Yes, and since we can solve additional subtraction from left to right, we can first do 11 minus three, which is going to give us eight. And next we can do eight plus six and reach our final answer off. 14. Great. Next. Let's take a look at 20 plus seven minus five and by quickly scanning, penned as we can see, we don't have any parentheses, exponents, multiplication or division. We do have addition and subtraction, and we couldn't do it from left to right. So first we dio 20 plus seven, which is going to give us 27. Next we subtract 27 minus five, which is going to give us a final answer of 22. Next, let's take a look at seven minus one, divided by two times five. Let's quickly scan AARP, MGA's indicator and weaken quickly. Spot that we have a parentheses. So we do that first seven minus one, which is going to give us six. And we can rewrite our problem as six divided by two times five and we can scan are pendants indicator again. But if you remember the third rule, you know when it comes to multiplication and division, you can solve it from left to right just how you read from left to right so we can do six divided by two, which is going to give us three. And for a final step, we do three times five, which is going to give us our final answer off 15. I hope you started to see that order of operation is not hard at all. Once you know the simple rules and if you have a hard time remembering the rules, you can know his resorts upend ass, which can also serve as a quick reminder of what should be done first or in the order. You should solve the problem. Let's try a few more just so you could get the hang of order of operation. Next, Let's take a look at nine plus five times two plus five squared. And this problem is a bit more challenging than the rest we've done so far. But no worries. Weakened scan our penthouse acronym and we can clearly see we don't have any parentheses. But we do have an exponents at the end. And this is where order of operation can get a big confusing because it recommends that you fully express your exponents. So you avoid further confusion so we can rewrite this problem as nine plus five times two plus 25 since five squared is 25. Next we can see that multiplication comes after exponents and PEM gas. So we do five times to which is going to give us 10 and we bring down the rest of our numbers in the same order they currently are. And all we have to do is addition, so we can solve this from left to right. So nine plus 10 is 19 and for our last step, we add 19 plus 25 which is going to give us 44. As you can see, this is fairly easy stuff. You just have to go step by step and take your time. And I can assure you the more you practice problems, the faster you'll be able to solve your order of operation problems. Far last problem. Let's take a look at six plus three times 10 squared, divided by two plus 10 immediately we see we have a parentheses, so we solve everything inside the parentheses first. And since we also have an exponents inside of our parentheses, we can start there to avoid confusion as we get deeper into solving our problem. So we can rewrite this problem as six plus three times 100 after we fully express our exponents because we know 10 squared is 100. Then we bring down the rest of our problem and we still have a parentheses. So we continue to solve everything inside the parentheses and looking upend as we can see that multiplication comes after exponents. So we multiply before we add. So we do three times 100 which is going to give us 300. We can rewrite our problem as six plus 300 still inside a parentheses, and bring down the restaurant problem to keep things in order. Next we add six plus 300 which is going to give us 306 As it's it's our last number inside of our parentheses. Weaken, just write it outside of a parentheses and now we have three or six divided by two plus 10 . And if you're not sure what to do first, we can look at Penn Days and we see that the vision is always soft before addition. So we divide 306 by two, which is going to give us 153 and far last step. We add 1 53 plus 10 and that's going to give us a final answer off 163. Great. I know we covered quite a few things in this video, and I recommend you watch it two or three times and practice with the quiz after this video and I can assure you, order of operation will become easier and easier as you do more problems. This is going to conclude this video on order of operation. I want to thank you for watching this first section and our next video we will be learning about factors and multiples. I want to thank you for watching. I look forward to seeing you in our next video. If you have any questions, please don't hesitate to ask and I'll be glad to assist you. Thanks.
11. Factors & Multiples: Hey, everyone, welcome back. At our last video, we learned about Order of Operations, which I hope you were able to get a firm grasp of, because it's a very important topic in algebra. In this video, we will be learning about factors and multiples, and I want to start by saying that this is one of the easier topics of pre algebra and algebra, and, it may seem like has nothing to do with algebra but in fact is one of the most important things you can learn, which will help you solve expressions as we get into other pre algebra and algebra topics. So first, let's learn what is a factor. A factor or factors are simply whole numbers he could multiply. To get a specific number or other numbers you can multiply together in order to get a given number. This may sound a bit complicated when you first read or hear this definition, but what this simply means is, let's say we have the number eight and remember, we said we can multiply the given number by itself to get the same number or other numbers to give us the given number as a result, so What are the factors of eight? If we multiply one times eight, that's going to give us eight. Therefore making one a factor off a. Now we ask ourselves, Is there any other number weekend multiplied by eight to give us eight? No. So we move on and ask ourselves, what other two numbers can we multiply to give us eight? Well, eight divided by two is four. So that means that we multiply two times, for that's going to give us eight. Now we ask ourselves again, Is there any two whole numbers, not decimal numbers? Remember, we said whole numbers that we can multiply to give us an answer of eight, and the answer is no. So here we can see we have all our multiples of eight, which is one to four and eight. And what we're supposed to learn from this is that the factors of the number eight are 124 and eight. And if during your test you're asked what are the factors of eight, you're basically being asked what to whole numbers? Can you multiply in order to get eight, or what number can you multiply by eight to get the number eight, and a good way to check if it's a multiple of a given number is to divide the given number by the factors, and your answer should be divided evenly with no remainder. So let's divide eight by one that's going to give us eight. Next. Let's take a look at eight. Divided by two that's going to give us four last but not least eight divided by four, which is going to give us too great. Let's look at another number. Six. First, you want to ask yourself the basic question of what can I multiply by six. That can give me six as a result, and also, what two numbers can I multiply in order to give me six? And this is where no, your multiplication table can come in handy, so we know six times one is going to give us six, therefore making one a factor of six. Next, we ask ourselves, Will other number multiply by six can give us six, and the answer is none because two is next after one and two times six would give us 12 so next will check to see if there are any other two numbers we can multiply by each other to give us six, and we know two times three gives up. Six. So two and the number three are also factors of six. Next, we ask ourselves, Are there any other two numbers that we can multiply toe? Arrive at six and the answer is no. And that's usually the case with small numbers. So we have all the factors of six. 123 and six Great. And so that will check. We divide our factors by six in order to see if they go into six evenly. So six divided by one a six. Next, we look at six divided by two, which is going to give us three and last, but not least six divided by three is going to give us, too. Let's look at another example, because that's of course, how we will get good at this is by looking at various problems and learning the steps to solve the problems over and over until it becomes second nature and barely requires any thinking to solve it. So next, let's take a look at 10. As you've probably noticed from the two previous examples, one is usually a factor because it gives you the given number, which in this case would be 10 times one is 10. Next we ask ourselves what two numbers can we multiply in order to get 10? We know till times five is 10. Therefore two and five are factors of 10. So here we can see we have on our factors of 10 because no other two numbers can be multiplied to get 10. And you can also, you can only multiply one by 10 in order to get 10. This can be a little tricky if you don't know your times table, but I can assure you the better you know, your multiplication table, the better. It will be able to identify all the factors of a given number and to make sure our factors of 10 are correct. Less divide. So 10 divided by one is going to give us 10 10. Divided by two is going to give us five and 10. Divided by five is going to give us too. And here we can see one of the reasons why I love math and you may have not noticed this pattern. But as we divide our factors of 10 you can see that it's all connected. For example, when we divided 10 by two, we got five. What we divided 10 by five. We got to, and I point this out because as we get into more complex algebra topics, you'll start to notice more and more patterns like this, which will only hope You understand math anyway, you never understood it before another. We've covered factors. I would like to talk about it closely Related topic multiples. What is a multiple? A multiple is basically when you multiply by a specific given number. For example, let's say were given the number five and you're asked, What are some of the multiples of five? You may have learned this when you were learning your multiplication table. All you have to do is count by five, for example, 5 10 15 20 25 30. So, as you can see, you start with the given number, which is five, and you increment by five each time you can't. Another reason this is called the multiple is because each increment you make by the given number is multiplied by the given number in order to get the multiple. So, for example, five times. One is five, five times two is 10 five times three is 15 five times four is 20 five times five is 25 and five times six is 30. And from all of these multiplication you can see one common multiple, which is five. You may ask yourself, what is the difference between a factor? Any multiple with a factor? There is a limited amount of numbers you can get with a given number and with multiples. The numbers are limited because you can keep multiplying by the given number to get your next multiple another you nobody multiple and a factor is let's do if you practice problems far. First problem. Let's find out what are the factors off 20. And as we've learned, one is always a factor because multiplied by the given number is going to give you the given number. So 20 times one is going to give you 20. Next we ask ourselves what two numbers can we multiply to get 20 while five times four gives us 20 making four and five factors of 20. Another factor of 20 would be two and 10 because two times 10 is 20 and here we have all our factors off 20. That's a double check. We divide the factors by 20. So 20 divided by one is going to give us 20 20. Divided by two is going to give us 10 20. Divided by four is going to give us five and 20. Divided by five is going to give us four. Great. Next. Let's take a look at the factors off 30 from previous examples. We know one as a factor because one times 30 gives us 30. Next, we ask ourselves what two numbers can I multiply in order to get 30? Well, 15 is half of 30 so two times 15 is going to give us 30 therefore making two and 15 factors of 30. Another factor would be three times 10 which also gives us 30 and last but not least five times six also gives us 30 therefore also making them factors of 30. And here we have all our factors of 30. Great next. I would like to practice with a few multiples before we conclude this video. So let's start with two. And as we learned, finding the multiples basically means to count or multiply by the given number. So next we have four says two times two is four. Next, we have six than eight and will only be doing five so we can stop at 10. And we didn't expand on this by seeing how we multiply, too, to arrive at our multiples of two. Next, let's look at multiples of 10. And just as before, all we have to do is multiplied by 10 each time in order to find our multiples of 10. So next would be 20 than 30 than 40 and 10 times five gives us 50 so we can stop here. And this is going to conclude this video on factors and multiples. In our next video, we will be learning all about prime numbers and why it's important to know what a prime number is and how toe identify prime numbers. I want to thank you for watching this video, and I look forward to see you in our next