Algebraic Expressions - Online Tutor, Practice Problems & Exam Prep

Algebraic expressions combine numbers, variables, and operations, with key components including coefficients, constants, and variables. Evaluating these expressions involves substituting values for variables and applying the order of operations (PEMDAS). Exponents represent repeated multiplication, simplifying expressions like \(x^3\) to \(x \cdot x \cdot x\). Simplifying involves combining like terms, which are terms with the same variable and degree. Understanding these concepts is essential for manipulating polynomials and solving equations effectively.

Introduction to Algebraic Expressions

Welcome back, everyone. In previous videos, we saw how to solve problems like 2 times 3 plus 5, which involved numbers and operations only. These are what we call numerical expressions. In this video, we're going to talk about algebraic expressions, which are a little bit different, and this is, unfortunately, the part of math where we're going to start to mix up numbers with letters of the alphabet. I know that sounds scary at first, but I promise I'm going to break it down for you, and I'm going to 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 2 times 3 plus 5 was numbers and operations only, in this case, where we have 2x + 5, 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, variables, and math operations. This variable here, this letter x, is just a letter that can represent any number. The idea here is that this x could be 3, but it also could be negative 2. It could even be 0, 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 2 that sort of sits in front of this x is called the coefficient. So a coefficient is just a number that goes, let's say, attached to a variable, and when you see it in front, it means that it's multiplying the variable. Unlike the x, this 2 doesn't change. It can't be a 3 or a negative 2 or a 0, 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 5, and this 5, unlike the 2, is a number that is without a variable. It's not attached to an x. And just like the 2, its value doesn't change. This 5 can't become a 2 or a negative 3 or something like that, and this is called a constant. Constants are usually seen at the end of your algebraic expressions, but that's basically all there is to it. 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 going to determine which of the following are expressions, and we're going to identify coefficients and constants. Let's get started.

In part a, remember, what we're looking for here is numbers, operations, and variables. So here I have a number, which is 4. 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 4 or is it the 8? This 4 is attached to this sort of square root symbol of x, but it goes in front and it multiplies, so this thing is going to be your coefficient. And this constant, or this 8 here that's by itself, is not attached to a variable. It's by itself, so this is going to be your constant.

So let's look at part b now, this 3 ( 14 + 5 ) / 6. So I definitely got numbers, and I definitely got operations like multiplication, addition, and division even. What I don't have here is a variable. So because I have no variable in this expression, it's actually not going to be an expression. This is just a good numerical expression. It's not algebraic.

So let's move on now to part c. Part c is 2 - 3xy. 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 2 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 do you think it's the 2 or the negative 3 or the 3 that's over here? Well, hopefully, you realize that the 3 is the one that's attached to the variable. So this is going to be your coefficient over here, and this 2 is going to 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 9 x = 18. I have a number over here, and I also have multiplication, so that's a symbol, and I also have a variable over here. So is this an algebraic expression? Well, it would be, except for this equals sign. 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, but this is actually just going to be an equation, so it is not an expression. Anyway, that's the basics. Let's keep moving on, and thanks for watching.

Evaluating Algebraic Expressions

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 3, negative 2, or even 0. The idea here is that just like with numbers, we're going to have to use operations, we're going to 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 3, and this is called evaluating an algebraic expression. When you evaluate an expression, what you're going to do is you're going to plug in those given values that the problem is telling you. You're going to plug those in for the variables, and then we're just going to use our order of operations. We're just going to use, first, remember, PEMDAS, and I've got it here just in case you need a little bit of a refresher. So the idea here is that \(2x + 5\), that x could mean anything. But in this particular problem, it's telling you that x is equal to 3. So all we have to do is wherever we see an x, we just replace it and plug in a 3. That's what evaluating an expression means.

Let's just jump right into this problem here. This \(2x+5\) if x is equal to 3, then I'm just going to replace the x with a 3. So this is \(2(3) + 5\). You're usually going to see when you plug in stuff or when you plug in numbers for variables, they get this little parenthesis here. And so that's all there is to it. This x is equal to 3. 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 2 and the 3, this becomes 6, and then \(6 + 5 = 11\). Alright? So this is how you evaluate an algebraic expression. Everywhere you see an x, just replace it with a 3. Alright?

So let's move on to the second part problem, now part b, which is a little bit more complicated. So we have negative and then we have \(2(8 - x)\) all divided by \(4x\), but the idea is the same. Everywhere we see an x, we just replace it with a 3. So this just becomes, \(-\frac{28-3}{4(3)}\). Notice how this x just became a 3, and this x on the bottom also just became a 3. Now we just have to use our order of operations. So first, we have to deal with the parentheses first. So we have negative, and this is going to be 2. Then we have 8 minus 3. This just becomes a 5. And then on the bottom, we're not going to do anything yet. This is going to be 4 times 3. 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 \(2 \times 5\) is just equal to, don’t forget the minus sign, \(-10\), and over here, \(4 \times 3\), this just becomes 12. Alright? So all we have to do here is just now simplify the fraction. We've seen how to do this before. This is \(-\frac{10}{12}\), which simplifies to \(-\frac{5}{6}\), and this is the answer to your problem. Alright? So that's what it means to evaluate an algebraic expression. Let me know if you have any questions, and thank you for watching.

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

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

• A. 43
• B. 28
• C. -2
• D. -22

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

Introduction to Exponents

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 going to do in this video is I'm going to show you that we have a special notation for writing this called exponent notation. So I'm going to show you that this 4 times itself a bunch of times can actually just be written as 45. That's what I'm going to 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 4 times itself 5 times, 4, 1, 2, 3, 4, 5. And so, basically, what this 4 represents is the 4 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 5 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 45, and we actually say it as it's 4 raised to the 5th power. Alright? So that's all an exponent is, is it just says this thing is multiplied by itself a bunch of times. Alright? Now in some cases, we want to have to condense, a bunch of, you know, numbers into a smaller format. We'll actually have to expand it out and see what all the multiplication is. So, for example, if I have something like x to the exponent of 3, that just means x×x×x. Right? So we can expand it and condense it as well. And, by the way, so the x the base here is just x, and the exponent is 3. And one of the other ways 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 exponent is if I have something 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, there's, like, a there's a bunch of a's here, then that just means I can say that this is an raised to the nth power. That's the general notation for this. Alright? So other than that, that's really all there is to it. So let's just go ahead and take a look at some problems now because now our algebraic expressions are going to involve some exponents, but, really, we're going to be doing the same thing. We're going to be evaluating expressions. We know exactly how to do that. So let's start with our first problem here. We have negative 3x to the 4th 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 2. So, for example, in this problem here, I have negative 3, and then I have, remember, I have to put a parenthesis here, 2 raised to the 4th 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 PEMDAS. PEMDAS here says that we always do exponents before we do things like multiplication, division, addition, and subtraction. Exponents are 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 3 and the 2 before they've done the exponent, and they'll do something like negative 6 to the 4th power. And this is wrong. This is wrong. Don't do this. If you do this, you're going to get the wrong answer. Alright? So just be very careful. So, really, what happens is you actually sort of have to take care of the 24 first. You have to do that before you do this multiplication. So this is really like 3 negative 3 times 2 times 2 times 2 times 2. That's what 24 means. It's just 2 multiplied by itself a bunch of times. And it might be helpful to write out all the multiplication because you may not know what 24 is off the top of your head, and that's fine because you could write this out, and 2 times 2 is just equal to 4, and then 2 times 2 is just equal to 4. So, really, this is just negative 3 and then 4 times 4. And that's a little bit easier to solve because we know 4 times 4 is just 16. So in other words, your final answer, negative 3 times 16, is actually just negative 48. And that's the answer that's how you evaluate an expression with an exponent in it. Alright? Now let's take a look at the second problem here. Here we have y2 plus 102. Alright? So remember, evaluating an expression just means that I'm going to replace a y with a 5. Some other words, I just replace the y with a 5 over here, 52 plus 102. Alright? Now remember, order of operation 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 5 times itself, 5 times 5 plus, and this is just 10 times 10. So remember, you kind of just do those two things first before you do the addition or subtraction. So in other words, the 5 times 5 is just 25. That's what this becomes. Plus the 10 times 10 is just a 100. So, therefore, your final answer is a 125. Alright? 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 2, so we just replace the x with a 2. And y equals 5, we just replace the y with a 5. So in other words, this just becomes this x to the third power. This actually just becomes 2 to the third power plus 4 times, and this just becomes y. So that's a 5, and so this is just going to be minus 7. Okay? 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 2 to the third power first before we do anything else. So in other words, what happens is 2 to the third power is really just 2 times 2 times 2 plus, and then we have 4 times negative 5. So 4 times negative 5. And then we have and then we have minus 7 on the outside here. Alright. So we have to do the multiplication before we do addition and subtraction. This actually ends up becoming 8 2 times 2 times 2, and this 4 times negative 5 actually becomes negative 20. So now because I'm doing addition and subtraction, I can actually just drop the parentheses over here. This is really just 8 minus 20. Alright? So 8 minus 20, but this is still in the parentheses, so you have to just drop that. So you you can't drop that. And then minus 7, so we have to do this thing first. And this 8 minus 20 is just negative 12. Negative 12, now we can drop the parentheses, minus 7, and this just becomes negative 19. Alright? 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.

Simplifying Algebraic Expressions

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 3, I just replace the x with 3 and so on with y. But some problems are not going to have you do that. In some problems, you're going to have to take a long, complicated expression, something that might look like this, and you're going to have to write it in a simpler form. And that's called simplifying an algebraic expression. It's what I'm going to show you how to do in this video. Basically, by the end of this, I'm going to show you a step-by-step process for how something like this expression actually just simplifies down to just a variable x. I'm going to 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 5 minus x plus 3y plus y. There's a minus sign here, a plus sign, and a plus sign. So all these four things here are parts of my expression. Those are all just terms, and we can see here that some terms are actually just numbers only, like 5. Some of them are variables only, like x and y, and then some of them are actually just combinations of numbers and variables, like 3y. Alright? All these things are terms. Now 2 of these things are more similar than others. And what do I mean by that? The 5 and the x aren't similar because one's a number and one's a variable. The x and the y are 2 different variables, but this 3y 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 going to minus an apple plus 3 bananas plus another banana. It's like you're talking about the same thing. So these things here are called like terms, the 3y 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 y² or something like that. Okay? 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 5 minus x. I can't combine those because they're not similar, but it's like I have 3 bananas and 1 banana, so I can combine that and just say, well, I just have 4 bananas. Alright? That's the whole thing is that you're just going to 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 going to simplify this algebraic expression here. We have 2x plus 3 plus 4 and then parentheses x plus 2. I'm going to simplify this. Remember, that means I want to reduce the number of terms. So the first thing you're going to have to do in this is actually kind of follow some order of operations. I see this 4 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 4 distributes into the x and the 2, and it just becomes 4x+8. We've seen that before, and I'll just rewrite the other terms over here. This is 2x. Alright. So now what I what I can see here is I have a term that has a 2x and a term that has a 4x, and then I have a term that was just a 3 and an 8. So I've got some stuff that is variables and some things that are numbers here. Alright? So that brings us to the second step. So we already distributed. The second step is you're going to 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 2x and the 4x so that they're side by side. So what I have to do is I have to get the 2x and I have to bring over the 4x, but I have to bring over the sign that's in front. So in other words, I have to bring over the 4x to bring that whole thing over and make sure that I'm keeping my signs correct. So then I have a plus 3, and then I have a plus 8 over here. Alright. So you're kind of just picking up these terms and repositioning them, and you can do that because everything's added here. Alright. 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 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 kinda like what we did up here. We add 3 bananas and 1 banana into 4 bananas. Now we just do the same exact thing. Right? So I could basically just say that this 2x and this 4x is like 2 apples and 4 apples. This basically just condenses down to 6x. Right? Something like that. So that's the idea here. So I can combine those like terms. This becomes 6x, and I can combine the 3 and the 8 because those are just numbers, and this ends up being 6x + 11. And that's as far as I can go. I can't add 6x and 11 because they're not like terms. It's like I'm adding 6 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 4 terms with parentheses and stuff like that, and we'll see that this actually just simplifies to a very simple expression with 2 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.

Simplify

$-3\left(5-x\right)+10-7x$

• A. $13x-22$
• B. $-4x-5$
• C. $-22x+13$
• D. $-8x-5$

Simplify

$-13+4x+x\left(9-x\right)$

• A. $-x^2+4x+9x-13$
• B. $-x^2+13x-13$
• C. $12x-13$
• D. $x^2+13x-13$

Simplify

$3x+14y-7\left(-x+2y\right)$

• A. $10x$
• B. $10x+16y$
• C. $10y$
• D. $17x-7y$

Here’s what students ask on this topic:

What is an algebraic expression?

An algebraic expression is a mathematical phrase that combines numbers, variables, and operations. For example, in the expression $\mathrm{2x}+5$, '2' is a coefficient, 'x' is a variable, and '5' is a constant. The variable can represent any number, and the expression can be evaluated by substituting specific values for the variable. Understanding algebraic expressions is fundamental for solving equations and manipulating polynomials.

How do you evaluate an algebraic expression?

To evaluate an algebraic expression, substitute the given values for the variables and then apply the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For example, to evaluate $\mathrm{2x}+5$ when $x=3$, substitute 3 for x: $23+5=6+5=11$.

What are coefficients and constants in algebraic expressions?

In algebraic expressions, a coefficient is a number that multiplies a variable. For example, in $\mathrm{3x}$, '3' is the coefficient. A constant is a number that stands alone without a variable, such as '5' in $\mathrm{3x}+5$. Coefficients and constants are essential for forming and simplifying algebraic expressions.

How do you simplify an algebraic expression?

To simplify an algebraic expression, combine like terms, which are terms with the same variable and exponent. For example, in $\mathrm{2x}+\mathrm{3x}+5+4$, combine $\mathrm{2x}$ and $\mathrm{3x}$ to get $\mathrm{5x}$, and combine $5$ and $4$ to get $9$. The simplified expression is $\mathrm{5x}+9$.

What is the role of exponents in algebraic expressions?

Exponents in algebraic expressions represent repeated multiplication of a base number or variable. For example, $x^3$ means $xxx$. Exponents simplify expressions by condensing repeated multiplication into a more compact form. When evaluating expressions with exponents, always perform the exponentiation before other operations, following the order of operations (PEMDAS).