What are exponents?

Exponents room numbers that have actually been multiplied by themselves. Because that instance, 3 · 3 · 3 · 3 could be composed as the exponent 34: the number 3 has been multiply by itself 4 times.

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Exponents are useful because they let united state write long numbers in a to rdoyourpartparks.orgce form. Because that instance, this number is very large:

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

But you could write the this method as one exponent:

1018

It likewise works for little numbers with countless decimal places. Because that instance, this number is very little but has plenty of digits:

.00000000000000001

It also could be composed as an exponent:

10-17

Scientists often use exponents to convey very huge numbers and also very tiny ones. You'll view them often in algebra problems too.

Understanding exponents

As you observed in the video, exponents are written prefer this: 43 (you'd read it together 4 come the 3rd power). Every exponents have two parts: the base, i m sorry is the number gift multiplied; and also the power, i m sorry is the number of times you main point the base.

*

Because our basic is 4 and our strength is 3, we’ll should multiply 4 by itself three times.

43 = 4 ⋅ 4 ⋅ 4 = 64

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

Occasionally, you can see the very same exponent written like this: 5^3. Don’t worry, it’s exactly the exact same number—the basic is the number to the left, and the power is the number to the right. Relying on the type of calculator girlfriend use—and especially if you’re utilizing the calculator on her phone or computer—you may need to input the exponent this method to calculate it.

Exponents to the first and 0th power

How would certainly you simplify these exponents?

71 70

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

Exponents through a strength of 1Any exponent v a power of 1 equals the base, therefore 51 is 5, 71 is 7, and also x1 is x.Exponents through a power of 0Any exponent with a strength of 0 amounts to 1, therefore 50 is 1, and also so is 70, x0, and also any various other exponent v a power of 0 you can think of.

Operations v exponents

How would certainly you deal with this problem?

22 ⋅ 23

If girlfriend think you need to solve the exponents first, climate multiply the resulting numbers, you’re right. (If friend weren’t sure, inspect out our lesson on the bespeak of operations).

How about this one?

x3 / x2

Or this one?

2x2 + 2x2

While girlfriend can’t precisely solve these difficulties without an ext information, you have the right to simplify them. In algebra, friend will regularly be request to perform calculations on exponents through variables together the base. Fortunately, it’s straightforward to add, subtract, multiply, and also divide these exponents.

Adding exponents

When you’re adding two exponents, you don’t include the actual powers—you include the bases. Because that instance, to simplify this expression, you would certainly just include the variables. You have two xs, which have the right to be composed as 2x. So, x2+x2 would be 2x2.

x2 + x2 = 2x2

How around this expression?

3y4 + 2y4

You're adding 3y to 2y. Due to the fact that 3 + 2 is 5, that method that 3y4 + 2y4 = 5y4.

3y4 + 2y4 = 5y4


You can have noticed the we only looked at troubles where the exponents us were including had the very same variable and also power. This is because you can only add exponents if their bases and exponents room exactly the same. So you can add these below since both terms have actually the exact same variable (r) and also the very same power (7):

4r7 + 9r7

You have the right to never add any of these together they’re written. This expression has actually variables through two different powers:

4r3 + 9r8

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

4r2 + 9s2

Subtracting exponents

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

5x2 - 4x2

5-4 is 1, for this reason if you stated 1x2, or just x2, you’re right. Remember, just like with including exponents, you can only subtract exponents v the same power and also base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying exponents is simple, however the way you do it could surprise you. To main point exponents, add the powers. For instance, take it this expression:

x3 ⋅ x4

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

x3 ⋅ x4 = x7

What about this expression?

3x2 ⋅ 2x6

The powers space 2 and 6, for this reason our simplified exponent will have actually a strength of 8. In this case, we’ll likewise need to main point the coefficients. The coefficients room 3 and 2. We must multiply these choose we would any kind of other numbers. 3⋅2 is 6, for this reason our streamlined answer is 6x8.

3x2 ⋅ 2x6 = 6x8

You can only simplify multiplied exponents with the exact same variable. Because that example, the expression 3x2⋅2x3⋅4y2 would certainly be streamlined to 24x5⋅y2. For an ext information, go to our Simplifying expression lesson.

Dividing exponents

Dividing index number is similar to multiplying them. Instead of adding 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 around this one?

10x4 / 2x2

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

Raising a power to a power

Sometimes you could see one equation like this:

(x5)3

An exponent on an additional exponent can seem confusing at first, but you currently have all the an abilities you need to simplify this expression. Remember, one exponent method that you're multiply the base by chin that many times. For example, 23 is 2⋅2⋅2. That means, we have the right to rewrite (x5)3 as:

x5⋅x5⋅x5

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

There's actually an even shorter way to leveling expressions prefer this. Take an additional look in ~ this equation:

(x5)3 = x15

Did you notice that 5⋅3 additionally equals 15? Remember, multiplication is the same as adding something much more than once. That method we have the right to think that 5+5+5, which is what we did earlier, together 5 times 3. Therefore, once you raise 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 much more example:

(3x8)4

First, we deserve to rewrite this as:

3x8⋅3x8⋅3x8⋅3x8

Remember in multiplication, bespeak does no matter. Therefore, we have the right to 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 very same as 34⋅x32.

Still confused about multiplying, dividing, or increasing exponents come a power? check out the video clip below to learn a trick because that remembering the rules: