Exponents

In this lesson on algebra exponents are covered. Get information and algebra exponents practice here.

Operations with exponents

How would you solve this problem?

22 ⋅ 23

If you think you should solve the exponents first, then multiply the resulting numbers, you’re right. (If you weren’t sure, check out our lesson on the order of operations).

How about this one?

x3 / x2

Or this one?

2x2 + 2x2

While you can’t exactly solve these problems without more information, you can simplify them. In algebra, you will often be asked to perform calculations on exponents with variables as the base. Fortunately, it’s easy to add, subtract, multiply, and divide these exponents.

Adding exponents

When you’re adding two exponents, you don’t add the actual powers—you add the bases. For instance, to simplify this expression, you would just add the variables. You have two xs, which can be written as 2x. So, x2+x2 would be 2x2.

x2 + x2 = 2x2

How about this expression?

3y4 + 2y4

You're adding 3y to 2y. Since 3 + 2 is 5, that means that 3y4 + 2y4 = 5y4.

3y4 + 2y4 = 5y4

You might have noticed that we only looked at problems where the exponents we were adding had the same variable and power. This is because you can only add exponents if their bases and exponents are exactly the same. So you can add these below because both terms have the same variable (r) and the same power (7):

4r7 + 9r7

You can never add any of these as they’re written. This expression has variables with two different powers:

4r3 + 9r8

This one has the same powers but different variables, so you can't add it either:

4r2 + 9s2

Subtracting exponents

Subtracting exponents works the same as adding them. For example, can you figure out how to simplify this expression?

5x2 - 4x2

5-4 is 1, so if you said 1x2, or simply x2, you’re right. Remember, just like with adding exponents, you can only subtract exponents with the same power and base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying exponents is simple, but the way you do it might surprise you. To multiply exponents, add the powers. For instance, take this expression:

x3 ⋅ x4

The powers are 3 and 4. Because 3 + 4 is 7, we can simplify this expression to x7.

x3 ⋅ x4 = x7

What about this expression?

3x2 ⋅ 2x6

The powers are 2 and 6, so our simplified exponent will have a power of 8. In this case, we’ll also need to multiply the coefficients. The coefficients are 3 and 2. We need to multiply these like we would any other numbers. 3⋅2 is 6, so our simplified answer is 6x8.

3x2 ⋅ 2x6 = 6x8

You can only simplify multiplied exponents with the same variable. For instance, you cannot simplify x2⋅y2.

Dividing exponents

Dividing exponents is similar to multiplying them. Instead of adding the powers, you subtract them. Take this expression:

x8 / x2

Because 8 - 2 is 6, we know that x8/x2 is x6.

x8 / x2 = x6

What about this one?

10x4 / 2x2

If you think the answer is 5x2, you’re right! 10 / 2 gives us a coefficient of 5, and subtracting the powers (4 - 2) means the power is 2.

Raising a power to a power

Sometimes you might see an equation like this:

(x5)3

An exponent on another exponent might seem confusing at first, but you already have all the skills you need to simplify this expression. Remember, an exponent means that you're multiplying the base by itself that many times. For example, 23 is 2⋅2⋅2. That means, we can rewrite (x5)3 as:

x5⋅x5⋅x5

To multiply exponents with the same base, simply add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15.

There's actually an even shorter way to simplify expressions like this. Take another look at this equation:

(x5)3 = x15

Did you notice that 5⋅3 also equals 15? Remember, multiplication is the same as adding something more than once. That means we can think of 5+5+5, which is what we did earlier, as 5 times 3. Therefore, when you raise a power to a power you can multiply the exponents.

Let's look at one more example:

(x6)4

Since 6⋅4 = 24, (x6)4 = x24

x24

Let's look at one more example:

(3x8)4

First, we can rewrite this as:

3x8⋅3x8⋅3x8⋅3x8

Remember in multiplication, order does not matter. Therefore, we can rewrite this again as:

3⋅3⋅3⋅3⋅x8⋅x8⋅x8⋅x8

Since 3⋅3⋅3⋅3 = 81 and x8⋅x8⋅x8⋅x8 = x32, our answer is:

81x32

Notice this would have also been the same as 34⋅x32.

Still confused about multiplying, dividing, or raising exponents to a power? Check out the video below to learn a trick for remembering the rules:

Assessment

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