What are exponents?

Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 can be composed as the exponent 34: the number 3 has been multiply by chin 4 times.

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Exponents space useful due to the fact that they let united state write lengthy numbers in a rrememberingsomer.comce form. For instance, this number is an extremely large:

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

But you could write the this way as one exponent:

1018

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

.00000000000000001

It likewise could be created as an exponent:

10-17

Scientists regularly use index number to convey very large numbers and very tiny ones. You'll view them frequently in algebra problems too.

Understanding exponents

As you saw in the video, exponents are written favor this: 43 (you'd check out it as 4 to the third power). All exponents have two parts: the base, i beg your pardon is the number gift multiplied; and the power, i beg your pardon is the variety of times you main point the base.

*

Because our basic is 4 and also our strength 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 same to 64, too.

Occasionally, you might see the same exponent written choose this: 5^3. Don’t worry, it’s specifically the very same number—the basic is the number come the left, and the power is the number to the right. Depending on the form of calculator friend use—and specifically if you’re making use of the calculator on her phone or computer—you may need to input the exponent this means to calculation 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. Also if you feel comfortable with other exponents, it’s not obvious how to calculation ones v powers the 1 and 0. Luckily, this exponents follow simple rules:

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

Operations v exponents

How would you deal with this problem?

22 ⋅ 23

If girlfriend think you must solve the exponents first, then multiply the result numbers, you’re right. (If friend weren’t sure, inspect out our lesson ~ above the stimulate of operations).

How around this one?

x3 / x2

Or this one?

2x2 + 2x2

While friend can’t precisely solve these problems without much more information, you have the right to simplify them. In algebra, friend will regularly be inquiry to carry out calculations on exponents through variables as the base. Fortunately, it’s straightforward to add, subtract, multiply, and divide this exponents.

Adding exponents

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

x2 + x2 = 2x2

How around this expression?

3y4 + 2y4

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

3y4 + 2y4 = 5y4


You can have noticed the we just looked at difficulties where the exponents us were including had the same variable and power. This is since you deserve to only add exponents if your bases and also exponents space exactly the same. So friend can include these below because both terms have actually the same variable (r) and the same power (7):

4r7 + 9r7

You have the right to never include any of these together they’re written. This expression has variables with two various powers:

4r3 + 9r8

This one has the same powers yet different variables, so friend can't include it either:

4r2 + 9s2

Subtracting exponents

Subtracting exponents works the very same as including them. Because that example, can you figure out exactly 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, as with with adding exponents, you deserve to only subtract exponents v the same power and also base.

5x2 - 4x2 = x2

Multiplying exponents

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

x3 ⋅ x4

The powers are 3 and 4. Since 3 + 4 is 7, we deserve to simplify this expression come x7.

x3 ⋅ x4 = x7

What around this expression?

3x2 ⋅ 2x6

The powers are 2 and also 6, therefore our streamlined exponent will have a strength of 8. In this case, we’ll likewise need to multiply the coefficients. The coefficients space 3 and also 2. We need to multiply these favor we would any other numbers. 3⋅2 is 6, for this reason our streamlined answer is 6x8.

3x2 ⋅ 2x6 = 6x8

You can only leveling multiplied exponents v the exact same variable. Because that example, the expression 3x2⋅2x3⋅4y2 would be simplified to 24x5⋅y2. For an ext information, walk to ours Simplifying expressions lesson.

Dividing exponents

Dividing index number is comparable to multiply them. Instead of including the powers, you subtract them. Take this expression:

x8 / x2

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

x8 / x2 = x6

What about this one?

10x4 / 2x2

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

Raising a power to a power

Sometimes you can see an equation choose this:

(x5)3

An exponent on another exponent might seem confusing at first, but you currently have all the an abilities you should simplify this expression. Remember, an exponent means that you're multiplying the base by itself that countless times. Because that example, 23 is 2⋅2⋅2. That means, we have the right to rewrite (x5)3 as:

x5⋅x5⋅x5

To main point exponents v 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 an additional look at this equation:

(x5)3 = x15

Did you notice that 5⋅3 additionally equals 15? Remember, multiplication is the very same as including something more than once. That method we deserve to think the 5+5+5, which is what we did earlier, together 5 time 3. Therefore, once you advanced a power come a power you can multiply the exponents.

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Let's look at one much more example:

(x6)4

Since 6⋅4 = 24, (x6)4 = x24

x24

Let's look in ~ one much more example:

(3x8)4

First, we deserve to rewrite this as:

3x8⋅3x8⋅3x8⋅3x8

Remember in multiplication, order does no matter. Therefore, we deserve 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 same as 34⋅x32.

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