What are exponents?

Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 3⁴: the number 3 has been multiplied by itself 4 times.

Exponents are useful because they let us write long numbers in a shortened form. For instance, this number is very large:

1,000,000,000,000,000,000

But you could write it this way as an exponent:

10¹⁸

It also works for small numbers with many decimal places. For instance, this number is very small but has many digits:

.00000000000000001

It also could be written as an exponent:

10^-17

Scientists often use exponents to convey very large numbers and very small ones. You'll see them often in algebra problems too.

Understanding exponents

As you saw in the video, exponents are written like this: 4³ (you'd read it as 4 to the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and the power, which is the number of times you multiply the base.

Because our base is 4 and our power is 3, we’ll need to multiply 4 by itself three times.

4³ = 4 ⋅ 4 ⋅ 4 = 64

Because 4 · 4 · 4 is 64, 4³ is equal to 64, too.

Occasionally, you might see the same exponent written like this: 5^3. Don’t worry, it’s exactly the same number—the base is the number to the left, and the power is the number to the right. Depending on the type of calculator you use—and especially if you’re using the calculator on your phone or computer—you may need to input the exponent this way to calculate it.

Exponents to the 1st and 0th power

How would you simplify these exponents?

7¹ 7⁰

Don’t feel bad if you’re confused. Even if you feel comfortable with other exponents, it’s not obvious how to calculate ones with powers of 1 and 0. Luckily, these exponents follow simple rules:

• Exponents with a power of 1
  Any exponent with a power of 1 equals the base, so 5¹ is 5, 7¹ is 7, and x¹ is x.
• Exponents with a power of 0
  Any exponent with a power of 0 equals 1, so 5⁰ is 1, and so is 7⁰, x⁰, and any other exponent with a power of 0 you can think of.

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

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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