Adding & Subtracting with Exponents

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  • 0:02 Exponent Basics
  • 2:39 Adding & Subtracting…
  • 4:34 Adding Exponents: Examples
  • 5:22 Subtracting Exponents:…
  • 6:12 Exponent of Zero
  • 7:10 Lesson Summary
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Lesson Transcript
Instructor: Vanessa Botts
Adding and subtracting with exponents can be quite easy once you know a few simple rules. In this lesson, you will learn about these rules and how to apply them any time you need to add or subtract with exponents.

Exponent Basics

Just like super-heroes, exponents have specific powers. They have the power to simplify repeated multiplication so we can save space! Okay, maybe their powers are not really that exciting, but they are certainly useful in math and science. We can perform all types of different mathematical calculations with exponents, but in this lesson, we will concentrate on only two: adding and subtracting with exponents.

Before starting to add and subtract with exponents, we should discuss a few things. What is an exponent? Exponents, or powers, correspond to the number of times a base is used as a factor. In other words, exponents indicate how many times a number should be multiplied by itself. For instance, when we see 5^3, it tells us that the base, 5, is multiplied by itself three times. In this case, the exponent is 3. If we write this out, it looks like this: 5^3 = 5 * 5 * 5, or 125.

The same applies when dealing with variables, or letters and symbols, like we will in this lesson. For example, x^3 would indicate that the x is multiplied three times, or x * x * x. The exponent, again, is 3.

Easy, right? Before moving on to adding and subtracting with exponents, we need to take a look at coefficients. A coefficient appears before a variable in an expression and is multiplied by that variable.

Let's take a look at the following expression: 2x^5. Hopefully you recognize that the base is x and the exponent is 5. But, what is different from the terms we saw earlier in the lesson? This term now also has a coefficient of 2.

Let's look at one more to make sure it makes sense. Can you identify the base, exponent, and coefficient?


This expression has a coefficient of 8, a base of x, and an exponent of 5.

By the way, if there is no number before the variable or symbol, then the coefficient will be 1. The same is true if you don't see an exponent. Therefore, the term y would have a coefficient of 1 and an exponent of 1; they just do not need to be written, which simplifies things a bit. Here is another example: 2x would indicate a coefficient of 2, a base of x, and an exponent of 1.

Got it? Now, we can move on to some simple rules so we can finally start adding and subtracting terms with exponents.

Adding and Subtracting with Exponents

When dealing with numbers only, we look at each expression, calculate, and then add or subtract as needed. The addition problem 2^2 + 3^3 becomes (2 * 2) + (3 * 3* 3). When we calculate each expression we get 4 + 27 = 31. Subtracting with exponents works the same way. 5^3 - 4^2 becomes (5 * 5 * 5) - (4 * 4). We then get that 125 - 16 = 109.

Different rules apply when dealing with variables. In order to add or subtract variables with exponents, you need to have like bases and like exponents, which means that the bases and exponents are the same. It does not matter if the bases are letters or even symbols; they need to be alike to be added or subtracted, and the exponents need to match.

For example, we can add and subtract the following expressions because they have like bases and like exponents:

2x^2 + 3x^2 = 5x^2

8y^3 - 5y^3 = 3y^3

Let's look at a different set of examples.

y^5 + x^5

2a^2 - a^3

Even though the exponents are the same, these cannot be added or subtracted because their bases or exponents are different.

Once we determine we do have the same bases and exponents, then all we need to do is add or subtract the coefficients. In other words, we add or subtract the coefficients together while keeping the bases and exponents exactly as they are.

Adding Exponents: Examples

Why don't we try to add the following?

2x^7 + 3x^7

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