Algebraic Expressions - Video Tutorials & Practice Problems

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Introduction to Algebraic Expressions

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Welcome back everyone. In previous videos, we saw how to solve problems like two times three plus five, which were numbers and operations only. These are what we call numerical expressions. In this video, we're gonna talk about algebraic expressions which are a little bit different. And this is unfortunately the part of math where we're gonna start to mix up numbers with letters of the alphabet. And I know that sounds scary at first, but I promise I'm gonna break it down for you and I'm gonna show you that algebraic and numerical expressions are actually very similar. So first let's get started with some basic vocabulary here. What is an algebraic expression? Well, whereas something like two times three plus five was numbers and operations only in this case where we have two X plus five, we have numbers operations and these things called variables. This letter X is called a variable. And so an algebraic expression is really just a combination of numbers and variables and math operations. All right, this variable here, this letter X is just a letter that can represent any number. The idea here is that this X could be three, but it also could be negative two, it could be even zero, something like that. So it stands in place for any number. And the idea is that this value varies. That's why we call it a variable. And usually the letter that we'll use in this course is the letter X. But later on, we'll see some other variables as well. Now, let's keep going here. This number two that sort of sits in front of this X is called the coefficient. So coefficient is just a number that goes and say attached to a variable. And when you see it in front, it means that it's multiplying a variable. All right now, unlike the X, this two doesn't change, it can't be a three or a negative two or a zero. So the value does not change for coefficients. And usually what you'll see is that coefficients go at the beginning of your algebraic expressions. Now, last but not least we have this five. And this five, unlike the two is a number that is without a variable, it's not attached to an X and just like the two, its value doesn't change this two. This five can't become a two or a negative three or something like that. And this is called a constant. All right. Now, uh constants are usually going to be seen at the end of your algebraic expressions. But that's basically all there is to it, right? It's numbers operations and variables. That's what makes up an algebraic expression. Let's get some more practice here. So we can see what kinds of things are expressions versus what aren't. So, in our example here, we're gonna determine which of the following are expressions and we're gonna identify coefficients and constants. All right, let's get started. So in part, a remember what we're looking for here is we're looking for numbers, operations and variables. We're gonna kind of go through that checklist here. So here I have a number which is four. I've also got an operation like a plus sign and I have a variable which is X. So is this an expression? It certainly is, it is an algebraic expression. Now, remember now we have to figure out coefficients and constants coefficients are numbers that multiply variables. So which one do you think it is, is it the four or is it the eight? Remember this four is attached to this sort of square root symbol of X? Um But it goes in front and then multiply. So this thing is gonna be your coefficients and this constant or this eight tier that's by itself is not, does not, you know, isn't attached to a variable, it's by itself. So this is gonna be your constant. All right. So let's look at part B now, this three parentheses, 14 plus five, divided by six. So I've definitely got numbers and I've definitely got operations like multiplication edition and division even what I don't have here. Is a variable. So because I have no variable in this expression, uh in this sort of thing over here, it's actually not going to be an expression. This is just like a numerical expression. It's not algebra. So let's move on now to part C, part C is two minus three XY. So we have numbers, we have symbols or operations like subtraction and even multiplication over here. And we've also got variables. In this case, we actually have two, we have X but we also have other letters like Y that can also be variables. So this definitely is an algebraic expression. Now, what's the coefficient? What's the constant? So what do you think it's the two or the negative three or the three that's over here? Well, hopefully realize that the three is the one that's attached to the variable. So this is gonna be your coefficient over here and this is gonna be your constant because it's off by itself without a variable. Now, you might be thinking, well, usually constants will go at the end and that's true. But this is actually a perfectly valid algebraic expression. Usually constants do go at the end and coefficients go at the beginning. But you can actually just see them in any order and this is perfectly fine. Last, but not least we have nine X equals 18. I have a number over here. And I also have multiplication. So that's a symbol and, and I also have a variable over here. So is this an algebraic expression? Well, it would be, except for this equals sign. And basically what you need to know here is that when expressions have an equals symbol between them, it actually forms what's called an equation. And all you need to know for right now is that equations are not actually considered algebraic expressions. We'll talk about them much later on. Um But this is actually just gonna be an equation. So it is not an expression anyway. So that's the basics. Uh Let's, let's keep moving on and thanks for watching.

2

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Evaluating Algebraic Expressions

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Now that we've seen the basics of algebraic expressions. Remember that this letter X here is variable is a letter that's used to represent any number. It could be three negative two or even zero. The idea here is that just like with numbers, we're gonna have to use operations, we're gonna have to add subtract multiply and divide variables when we're given their exact values. So the idea here is that some problems will actually tell us what X is equal to and it'll ask us to calculate something like it'll say X is equal to three. And this is called evaluating an algebraic expression. When you evaluate an expression, what you're gonna do is you're gonna plug in those given values that the problem is telling you you plug those in for the variables and then we're just gonna use our order of operations we're just gonna use or remember pem Ds and I've got it here just in case in a little bit of a refresher. So the idea here is that two X plus five that X could mean anything. But in this particular problem, it's telling you that X is equal to three. So all we have to do is wherever we see an X, we just replace it and plug in a three. That's what evaluating an expression means. Let's just jump right into this problem here. This two X plus five, if X is equal to three, then I'm just gonna replace the X with a three. So this is two parentheses, three plus five. You're usually gonna see when you plug in stuff or when you plug in numbers for variables, they get this little parentheses here. Um And so that's all there is to it. This X is equal to three. Now, we just have to use our order of operations. So we have to deal with parentheses first. And technically, there's a parenthesis here, but there's just one number inside of it. So it kind of just goes away, there's no exponents and then we've got multiplication and addition. So first, we have to multiply and divide before we add and subtract. And so when we multiply this two and the three, this becomes six and then six plus five is equal to 11. All right. So this is how you evaluate an algebraic expression everywhere. You see X just replace it with a three. All right. So let's move on to the second part problem. Now, part B which is a little bit more complicated. Um So we have negative and then we have two parentheses eight minus X all divided by four X. But the idea is the same everywhere we see an X, we just replace it with a three. So this just becomes a negative 28 minus three, divided by four. And then parentheses three, notice how this X just became a three and this X in the bottom also just became a three. Now we just have to use our order of operations. So first, we have to do with the parentheses first. So we have negative and this is gonna be two. then we have eight minus three. This just becomes a five. And then on the bottom, we're not gonna do anything yet. This is gonna be four times three. And now we just have a bunch of multiplication and division. There is no addition and subtraction in this problem. So all we have to do is we just go left to right and then top down right. So two times five is just equal to, don't forget the minus sign negative 10 or sorry, 10 over here. And on the bottom we have four times three and this just becomes 12, right. So all we have to do here is just now simplify the fraction and we've seen how to do this before. This is negative 5/6 and this is the answer to your problem. All right. So that's what it means to evaluate an algebraic expression. Let me know if you have any questions and thank you for watching.

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Problem

Problem

Evaluate the algebraic expression when $x=4$ and $y=-5$.

$2y-x\left(3+y\right)$

A

43

B

28

C

-2

D

-22

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Problem

Problem

Evaluate the algebraic expression when $x=-3$ and $y=2$.

$x\left(20-15y\right)-\left|2x+y\right|$

A

26

B

-21

C

-94

D

34

5

concept

Introduction to Exponents

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6m

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Everyone. So a lot of times in our algebraic expressions, we'll see the same thing, the number or a variable that gets multiplied by itself over and over again. And this is super inefficient and annoying to have to write out what I'm gonna do in this video is I'm gonna show you that we have a special notation for writing this called exponent notation. So I'm gonna show you that this four times itself a bunch of times can actually just be written as four with a little five on top. That's what I'm gonna show you in this video. Just exponents and expressions. Let's go ahead and get started here. So basically we use exponents to represent repeated multiplication. So for example, I had four times itself five times 412345. So basically what with this four represents is the four represents the base. It's the number or in some cases it could be a variable. So number or variable that's being multiplied a bunch of times and it's raised or sorry and it is multiplied five times. That's what we call the exponent or the power. It's basically the number of times that base is being multiplied. So we write this as four with a little five on top of it. And we actually say it as it or raised to the fifth power. All right. So that's all the exponent is, is, it just says this thing is multiplied by itself a bunch of times. All right. Now, in some cases, we want to have to condense a bunch of, you know, uh numbers into a smaller format. We'll actually have to expand it out and to see what all the, all the multiplication is. So for example, if I have something like X to the exponent of three, that just means X times X times X, right? So we can expand it and condense it as well. Uh And by the way, so the X, the base here is just X and the exponent is three. And one of the other ways that you might hear that is you might hear something like X cubed. That's what that third power means. So basically, the general form of any exponents is if I have some thing, a a is just a generic letter, it could be a number or a variable that's multiplied by itself a bunch of times. So in other words, it's like a dot dot dot There's a bunch, but there's a bunch of A's here. Then that just means I can say that this is a raised to the nth power. That's the general notation for this. All right. So other than that, that's really all there is to it. Um So let's just go ahead and take a look at some problems now because now our algebraic expressions are gonna involve some exponents. But really we're gonna be doing the same thing. We're gonna be evaluating expressions, we know exactly how to do that. So let's start with our first problem here. We have negative three X to the fourth power. And if we want to evaluate this algebraic expression, remember we just replace letters for numbers every time I see an X, I replace it with a two. So for example, in this problem here, I have negative three and then I have remember I have to put a parentheses here. Two raised to the fourth power. Now what's really important about exponents is and something you should always be cautious about is you always want to evaluate exponents before you do other operations. This is something that a lot of students will forget. But always just remember Pem Das pdas here says that we always do exponents before we do things like multiplication, division, addition and subtraction exponents is always the second thing that you do. So a lot of students, what they'll do is they'll take something like this and they'll multiply the three and the two before they've done the exponent and they'll do something like negative six to the fourth power and this is wrong, this is wrong. Don't do this. If you do this, you're gonna get the wrong answer. All right. So just be very careful. So really what happens is you actually sort of have to take care of the two race of the fourth power first. You have to do that before you do this multiplication. So this is really like three, negative, three times, two times, two times, two times two. That's what two to the fourth power means, it's just two multiplied by itself a bunch of times. Um And it might be helpful to write out all the multiplication because you may not know what two to the fourth power is off the top of your head. And that's fine because you, because you could write us out and two times two is just equal to four and then two times two is just equal to four. So really this is just negative three and then four times four. And that's a little bit easier to solve because we know four times four is just 16. So in other words, your final answer negative three times 16 is actually just negative 48. And that's the answer. That's how you evaluate an expression with an exponent in it. All right. Now, let's take a look at the second problem here. Here we have Y squared plus 10 squared. All right. So remember evaluating an expression just means that I'm gonna replace A Y with a five. So in other words, I just replace the Y with a five over here five squared plus 10 squared. All right. Now, remember order of operations says we have to do the exponents before we do anything else like addition or subtraction. So first take care of the exponents, this is really just five times itself five times five plus and this is just 10 times 10. So remember you kind of just do those two things first before you do uh the addition or subtraction. So in other words, the five times five is just 25. That's, this becomes plus the 10 times 10 is just 100. So therefore, your final answer is 100 and 25. All right. So that is the answer. Now, last but not least we have, you could have expressions involving multiple exponents and even multiple variables. So let's take a look at this one here here. Remember X equals two. So we just replace the X with a two and Y equals five, we just replace the Y with a five. In other words, this just becomes this X to the third power. This actually just becomes two to the third power plus four times. And this just becomes Y. Um So that's a five. And so this is just gonna be minus seven. OK? So remember we have to do the parentheses and we have to do the exponents first. In other words, we have to take a look at this before we can do the addition or subtraction and we have to do the exponents before we do the addition. So in other words, we have to take care of this two to the third power first before we do anything else. So in other words, what happens is two to the third power is really just two times, two times two plus and then we have four times negative five, so four times negative five and then we have, and then we have minus seven on the outside here. All right. So we have to do the multiplication before we do addition subtraction. This actually ends up becoming 82 times two times two. And this four times negative five actually becomes negative 20. So now because I'm doing addition subtraction, I can actually just drop the parentheses over here. This is really just eight minus 20. All right. So eight minus 20. Uh but this is still in the parentheses. So you have to just drop that. So you, you can't drop that and then minus seven. So we have to do this thing first and this eight minus 20 is just negative 1212. Now you can drop the parentheses uh minus seven and this just becomes negative 19. All right. So this whole expression here evaluates to negative 19, just plugging in a bunch of letters for numbers and that's how to deal with exponents and expressions. Let me know if you have any questions. Let's move on to the next video

6

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Simplifying Algebraic Expressions

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5m

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Welcome back, everyone. In the last couple of videos, we saw how to evaluate an algebraic expression. I could evaluate an expression like this by basically replacing letters with numbers. So if X was three, I just replace the X with three and so on with why? But some problems are not gonna have you do that in some problems, you're gonna have to take a long complicated expression, something that might look like this and you're gonna have to write it in a simpler form and that's called a simplifying an algebraic expression. It's what I'm gonna show you how to do in this video. Basically by the end of this, I'm gonna show you a step by step process or how something like this expression actually just simplifies down to just a variable X. I'm gonna show you exactly how that works. Let's get started. The idea here is that we can take a long expression and write it in a simpler form and that just comes down to reducing the number of terms. So let's talk about what a term is. A term is basically just a part or a thing in your expression that's separated by A plus or a minus sign. So for example, we have five minus X plus three Y plus Y, there's a minus sign here. A plus sign and A plus sign. So all these four things here are these parts of my expression, those are all just terms and we can see here that some terms are actually just numbers only like five. Some of them are variables only like X and Y. And then some of them are actually just combinations of numbers and variables like three Y. All right. All these things are terms. Now two of these things are more similar than others. And what do I mean by that? The five and the X aren't similar because one's a number and one's a variable, the X and the Y are two different variables. But this three Y and this one Y over here, those are similar. And the way I like to think about this is you can imagine an X is kind of like an apple and a Y is kind of like a banana. This expression is saying you're gonna minus an apple plus three bananas plus another banana. It's like you're talking about the same thing. So these things here are called like terms the three Y and the Y and basically like terms are just terms that have the same variable, they both have Y to the same exponent or the same power, right? Not one's not Y squared or something like that. OK. So the whole idea is that I can take these things and because they're like terms, I can combine them. So basically what this expression becomes is it becomes five minus X. I can't combine those because they're not similar. But it's like I have three bananas and one banana. So I can combine that and just say, well, I just have four bananas. All right, that's the whole thing is that you're just gonna be combining these like terms. Now let me show you a step by step process for how to do that. Let's just get into our example. So I could show you how this works. So we're gonna simplify this algebraic expression here. We have two X plus three plus four and then parentheses X plus two. I'm gonna simplify this. Remember that means I want to reduce the number of terms. So the first thing you're gonna have to do in this is actually kind of follow some order of operations. I see this four that's on the outside of a parenthesis. So the first thing you want to do is you want to distribute constants and variables into parentheses. If you have any, you kind of have to like expand this expression before you can start collapsing and reducing it. So that's what you have to do. First, the four distributes into the X and the two and it just becomes four X plus eight. We've seen that before and I'll just rewrite the other terms over here. Um This is two X. All right. So now what I, what I can see here is I have a term that has a two X and a term that has a four X and then I have a term that was just a three and an eight. So I've got some stuff that are variables and some things that are numbers here. All right. So that brings us to the second step. Um So we already distributed. The second step is you're gonna group together the like terms and the way you group them together is you just write them next to each other. So what do I mean by this? I want to basically write the two X and the four X so that they're side by side. So what I have to do is I have to get the two X and I have to bring over the four X, but I have to bring over the sign that's in front. So in other words, I have to bring over the four X to bring that whole thing over and make sure that I'm keeping my signs correct. Um So then I have a plus three and then I have a plus eight over here. All right. So you're kind of just picking up these terms and repositioning them and you can do that because everything is added here. All right. So that's done. So notice how we have now the terms that are similar to each other next to each other. So that's, that's grouping. Now, that brings us to the last step, which is just combining like terms. And the way we combine like terms is just by adding and subtracting. It's kind of like what we did up here, we add three bananas and one banana into four bananas. Now we just do the same exact thing, right? So I could basically just say that this two X and this four X is like two apples and four apples. This basically just condenses down to six apples, right? Something like that. So that's the idea here. So I can combine those like terms, this becomes six X and I can combine the three and the eight because those are just numbers and this ends up being six X plus 11. And that's as far as I can go, I can't add six X and 11 because they're not like terms. It's like I'm adding six apples to something that isn't an apple. So that is as far as you can go. And this is your simplified expression. This is how I take something that's four terms with parentheses and stuff like that. And we'll see that this actually just simplifies to a very simple expression with two terms. That's the whole thing guys. So let me know if you have any questions and I'll see you in the next video