What room exponents?

Exponents space numbers that have been multiply by themselves. Because that instance, 3 · 3 · 3 · 3 could be created as the exponent 34: the number 3 has been multiply by chin 4 times.

You are watching: 4 to the what power equals 64

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

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

But you could write it this way as one exponent:

1018

It additionally works for little numbers with many decimal places. For instance, this number is very tiny but has numerous digits:

.00000000000000001

It also could be written as an exponent:

10-17

Scientists often use exponents to convey very big numbers and very little ones. You'll check out them frequently in algebra troubles too.

Understanding exponents

As you witnessed in the video, exponents are written choose this: 43 (you'd read it as 4 come the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and also the power, i m sorry is the variety of times you multiply the base.

*

Because our base is 4 and also our strength is 3, we’ll must multiply 4 by chin three times.

43 = 4 ⋅ 4 ⋅ 4 = 64

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

Occasionally, you can see the same exponent written favor this: 5^3. Don’t worry, it’s precisely the very same number—the basic is the number come the left, and the strength is the number to the right. Depending upon the form of calculator girlfriend use—and specifically if you’re utilizing the calculator on your phone or computer—you may need to input the exponent this way to calculate it.

Exponents come the 1st and 0th power

How would you simplify these exponents?

71 70

Don’t feel bad if you’re confused. Also if you feeling comfortable with other exponents, it’s not noticeable how to calculation ones with powers of 1 and also 0. Luckily, these exponents follow basic rules:

Exponents with a strength of 1Any exponent with a power of 1 amounts to the base, for this reason 51 is 5, 71 is 7, and x1 is x.Exponents with a power of 0Any exponent with a strength of 0 amounts to 1, so 50 is 1, and also so is 70, x0, and any other exponent v a strength of 0 you deserve to think of.

Operations v exponents

How would certainly you deal with this problem?

22 ⋅ 23

If girlfriend think you need to solve the index number first, then multiply the resulting numbers, you’re right. (If you weren’t sure, examine out our lesson ~ above the bespeak of operations).

How around this one?

x3 / x2

Or this one?

2x2 + 2x2

While you can’t specifically solve these troubles without much more information, you have the right to simplify them. In algebra, girlfriend will frequently be inquiry to execute calculations ~ above exponents v variables together the base. Fortunately, it’s basic to add, subtract, multiply, and divide these exponents.

Adding exponents

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

x2 + x2 = 2x2

How around this expression?

3y4 + 2y4

You're including 3y come 2y. Due to the fact that 3 + 2 is 5, that way that 3y4 + 2y4 = 5y4.

3y4 + 2y4 = 5y4


You can have noticed the we only looked at troubles where the exponents we were adding had the exact same variable and power. This is due to the fact that you can only include exponents if your bases and exponents are exactly the same. So you can include these below due to the fact that both terms have actually the exact same variable (r) and the exact same power (7):

4r7 + 9r7

You deserve to never include any the these as they’re written. This expression has variables with two different powers:

4r3 + 9r8

This one has actually the exact same powers however different variables, so girlfriend can't include it either:

4r2 + 9s2

Subtracting exponents

Subtracting exponents works the same as including them. For example, deserve to you number out exactly how to leveling this expression?

5x2 - 4x2

5-4 is 1, therefore if you claimed 1x2, or simply x2, you’re right. Remember, similar to with adding exponents, you deserve to only subtract exponents with the same power and also base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying index number is simple, but the method you carry out it might surprise you. To multiply exponents, add the powers. For instance, take it this expression:

x3 ⋅ x4

The powers are 3 and 4. Due to the fact that 3 + 4 is 7, we deserve to simplify this expression to x7.

x3 ⋅ x4 = x7

What about this expression?

3x2 ⋅ 2x6

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

3x2 ⋅ 2x6 = 6x8

You have the right to only leveling multiplied exponents with the same variable. Because that example, the expression 3x2⋅2x3⋅4y2 would certainly be simplified to 24x5⋅y2. For more information, go to our Simplifying expressions lesson.

Dividing exponents

Dividing exponents is similar to multiply them. Instead of including the powers, you subtract them. Take it this expression:

x8 / x2

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

x8 / x2 = x6

What about this one?

10x4 / 2x2

If girlfriend think the prize is 5x2, you’re right! 10 / 2 provides us a coefficient that 5, and subtracting the strength (4 - 2) way the power is 2.

Raising a strength to a power

Sometimes you can see an equation choose this:

(x5)3

An exponent on one more exponent can seem confusing in ~ first, however you already have every the skills you must simplify this expression. Remember, an exponent means that you're multiply the base by chin that countless times. For example, 23 is 2⋅2⋅2. That means, we have the right to rewrite (x5)3 as:

x5⋅x5⋅x5

To main point exponents through the same base, just add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15.

There's actually an even shorter method to simplify expressions like this. Take one more look at this equation:

(x5)3 = x15

Did you notification that 5⋅3 also equals 15? Remember, multiplication is the exact same as including something much more than once. That means we deserve to think that 5+5+5, which is what us did earlier, as 5 time 3. Therefore, when you advanced a power come a power you have the right to multiply the exponents.

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Let's look in ~ one much more example:

(x6)4

Since 6⋅4 = 24, (x6)4 = x24

x24

Let's look at one an ext example:

(3x8)4

First, we deserve to 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 additionally been the same as 34⋅x32.

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