If x

     

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.

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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 và 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 khổng lồ the 3rd power). All exponents have two parts: the base, which is the number being multiplied; & the power, which is the number of times you multiply the base.

*

Because our base is 4 & our power is 3, we’ll need khổng lồ 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, và the power is the number khổng lồ 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 đầu vào the exponent this way to lớn calculate it.

Exponents lớn 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 lớn calculate ones with powers of 1 & 0. Luckily, these exponents follow simple rules:

Exponents with a nguồn of 1Any exponent with a nguồn of 1 equals the base, so 51 is 5, 71 is 7, and x1 is x.Exponents with a power of 0Any exponent with a nguồn of 0 equals 1, so 50 is 1, and so is 70, x0, và 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 khổng lồ add, subtract, multiply, & divide these exponents.

Adding exponents

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

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x2 + x2 = 2x2

How about this expression?

3y4 + 2y4

You're adding 3y to lớn 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 địa chỉ exponents if their bases & exponents are exactly the same. So you can địa chỉ cửa hàng these below because both terms have the same variable (r) and the same power nguồn (7):

4r7 + 9r7

You can never địa chỉ 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 địa chỉ 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 lượt thích with adding exponents, you can only subtract exponents with the same power & base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying exponents is simple, but the way you vị it might surprise you. Lớn 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 lớn 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 khổng lồ multiply these lượt thích 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 example, the expression 3x2⋅2x3⋅4y2 would be simplified lớn 24x5⋅y2. For more information, go to lớn our Simplifying Expressions lesson.

Dividing exponents

Dividing exponents is similar khổng lồ 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, và subtracting the powers (4 - 2) means the power is 2.

Raising a power to lớn 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

lớn 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 lượt thích 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 khổng lồ a power you can multiply the exponents.

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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 & 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 lớn a power? kiểm tra out the đoạn clip below lớn learn a trick for remembering the rules: