Combining Like Terms with Exponents — Rules & Examples

(Video) Adding and Subtracting Exponents

by Mometrix Academy

In this video she goes over what the parts of an exponent are and what you need in order to be able to add or subtract two exponents.

$\large{\text{Summary}}$

Recall the parts of an exponent. $$4x^2$$ The $4$ is called the coefficient. The $x$ is called the base. The $2$ is called the exponent.

To combine exponents, add or subtract them, we need two conditions to be met:

1. Same base
2. Same exponent

If either condition fails, we cannot combine the exponents.

Example 1

Let's look at an example: $$9x^2+4x^3$$ These terms cannot be combined because they don't have the same exponent part. The first term has a $^2$ and the second term has a $^3$. As a result, we cannot simplify this expression any more.

Example 2

$$5x^2-12y^2$$

Now, they have the same exponent but have different bases. The first term has an $x$ and the second term has a $y$. Therefore, we cannot combine them.

Example 3

$$5x^3+9x^3$$

These terms have the same base, $x$, and the same exponent, $^3$, so they can be combined. But how?

To add terms with the same base and exponent, just add the coefficients. In our example, this becomes: $$5x^3+9x^3=(5+9)x^3=\boxed{14x^3}$$

This is not to be confused with the product rule. That rule also involves manipulating exponents. However, it is for multiplying exponents together. Right now we are adding them instead.

Example 4

Sometimes, you'll have lots of terms. To simplify, you combine as many groupings of terms as you can. Sometimes we'll want to rearrange the terms to make this easier. We can do this because of the commutative property of addition. We must just carry around negative signs with us.

$$2x^2+5-4x^3+7x^3-x^2$$

We have two terms with $x^2$, two terms with $x^3$, and one term with neither. Therefore, to simplify, we can combine the $x^2$ terms and combine the $x^3$ terms. $$\begin{align} &{\color{#29A603}{2x^2}}+5\,{\color{#8925AE}{-\,4x^3}}+{\color{#8925AE}{7x^3}}\,{\color{#29A603}{-\,x^2}}\\ &={\color{#29A603}{(2-1)x^2}}+{\color{#8925AE}{-4+7}x^3}+5\\ &=x^2+3x^3+5 \end{align}$$ Finally, it's common practice to arrange the terms from largest exponent to smallest. $$=3x^3+x^2+5$$

This is called standard form.

You can add and subtract cubes with the same base

You can add and subtract cubed terms from each other as long as they have the same base (the number that the exponent is attached to).

Think of cubed numbers as blocks (or... literal cubes): you can only stack the ones that match, since we have no idea how the ones don't match compare in size and won't know if the stack will be stable.

Square everything, and then remember the exponent properties

When you're trying to simplify an expression with squared terms in it, the best thing to do is to square as much as you can, and then use the exponent properties to tidy up. For example, let's say we need to simplify the expression $(4x)^2 + x^2$.

Since I know I can square $4x$, I want to do that first. Don't worry that you're squaring a number and a variable.

$$\begin{align} (4x)^2\\ = (4^2)(x^2)\\ = 16x^2 \end{align}$$

So, we can rewrite our equation as $16x^2 + x^2$.

Now, notice that both terms include the same variable, which is being taken to the same power. Since that means they have the same value, we can add them together.

$$\begin{align} &16x^2 + x^2\\ &= 16x^2 + 1x^2\\ &= 17x^2 \end{align}$$

So, condensing our steps, we get: $$\begin{align} &(4x)^2 + x^2\\ &=16x^2 + x^2\\ &= 17x^2 \end{align}$$

Combine terms with the same variable and the same exponent

Remember that when you combine like terms, you combine the terms with the exact same variable by adding them or subtracting them, depending on the operation they have attached to them.

Terms with exponents work exactly the same way! You can only combine terms with the exact same variable and the exact same exponent, so you can only combine squared terms with squared terms, cubed terms with cubed terms, etc.

No, we can't combine any terms in $2x + 5y - 2x^2$. There are two terms with $x$ as their variable, but one $x$ is on its own and the other is squared ($x^2$). Since they're being raised to different powers, we can't combine them.

The $x$ and the $y$ are the same power, but they're totally different variables so we can't combine them either.

$$\color{#8925AE}{x^2} \color{#29A603}{-3x^3 + 2x^3} \color{#8925AE}{-7x^2}$$

$x^2$ and $-7x^2$ share exactly the same exponent and variable, so they can be combined into $-6x^2$ by just subtracting their coefficients. $-3x^3$ and $2x^3$ also share exactly the same exponent and variable, so they can be combined to become $-x^3$. Then, we just line up the terms we've combined into our new expression: $$\color{#8925AE}{-6x^2} \color{#29A603}{-x^3}$$