What room exponents?

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

You are watching: This is the power to which a number is raised, or the number of times it is multiplied by itself.

Exponents space useful since they let us write long numbers in a to rhistoricsweetsballroom.comce form. Because that instance, this number is really large:

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

But you can write the this method as an exponent:

1018

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

.00000000000000001

It additionally could be written as an exponent:

10-17

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

Understanding exponents

As you saw in the video, exponents space written choose this: 43 (you'd check out it together 4 come the 3rd power). Every exponents have actually two parts: the base, i beg your pardon is the number gift multiplied; and the power, i m sorry is the number of times you multiply the base.

*

Because our base is 4 and our power is 3, we’ll have 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 exact same exponent written prefer this: 5^3. Don’t worry, it’s exactly the same number—the base is the number come the left, and the strength is the number come the right. Depending on the kind of calculator girlfriend use—and especially if you’re using the calculator on her phone or computer—you might need come input the exponent this way to calculate it.

Exponents to the 1st and 0th power

How would you leveling these exponents?

71 70

Don’t feel negative if you’re confused. Even if you feeling comfortable with other exponents, it’s not apparent how to calculate ones through powers of 1 and also 0. Luckily, these exponents follow straightforward rules:

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

Operations v exponents

How would you settle this problem?

22 ⋅ 23

If you think you need to solve the exponents first, then multiply the result numbers, you’re right. (If friend weren’t sure, examine out our lesson on the order of operations).

How around this one?

x3 / x2

Or this one?

2x2 + 2x2

While girlfriend can’t specifically solve these difficulties without much more information, you deserve to simplify them. In algebra, friend will often be asked to execute calculations on exponents v variables together the base. Fortunately, it’s easy to add, subtract, multiply, and divide these exponents.

Adding exponents

When you’re adding two exponents, girlfriend don’t include the yes, really powers—you include the bases. For instance, to leveling 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 about this expression?

3y4 + 2y4

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

3y4 + 2y4 = 5y4


You could have noticed that we just looked at troubles where the exponents us were adding had the same variable and power. This is due to the fact that you deserve to only include exponents if their bases and also exponents are exactly the same. So friend can add these below due to the fact that both terms have the very same variable (r) and also the very same power (7):

4r7 + 9r7

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

4r3 + 9r8

This one has actually the same powers yet different variables, so girlfriend can't add it either:

4r2 + 9s2

Subtracting exponents

Subtracting exponents works the very same as including them. Because that example, can you figure out just how to leveling this expression?

5x2 - 4x2

5-4 is 1, so if you claimed 1x2, or just x2, you’re right. Remember, as with with adding exponents, you have the right to only subtract exponents v the same power and base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying exponents is simple, but the means you execute it could surprise you. To main point exponents, add the powers. For instance, take it this expression:

x3 ⋅ x4

The powers space 3 and also 4. Since 3 + 4 is 7, we have the right to simplify this expression to x7.

x3 ⋅ x4 = x7

What around this expression?

3x2 ⋅ 2x6

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

3x2 ⋅ 2x6 = 6x8

You deserve to only leveling multiplied exponents through the exact same variable. Because that example, the expression 3x2⋅2x3⋅4y2 would be streamlined to 24x5⋅y2. For an ext information, go to ours Simplifying expression lesson.

Dividing exponents

Dividing exponents is similar to multiply them. Rather 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 you think the answer is 5x2, you’re right! 10 / 2 offers us a coefficient of 5, and also subtracting the strength (4 - 2) way the power is 2.

Raising a strength to a power

Sometimes you might see one equation like this:

(x5)3

An exponent on an additional exponent might seem confusing at first, but you already have all the an abilities you need to simplify this expression. Remember, one exponent way that you're multiply the base by itself that numerous times. Because that example, 23 is 2⋅2⋅2. The means, we deserve to rewrite (x5)3 as:

x5⋅x5⋅x5

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

There's actually an even shorter method to simplify expressions choose 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 adding something much more than once. That means we have the right to think the 5+5+5, i m sorry is what we did earlier, together 5 time 3. Therefore, when you progressive a power to a power you deserve to multiply the exponents.

See more: ' One Thousand Dollars By O Henry, One Thousand Dollars Summary

Let's look in ~ one an ext example:

(x6)4

Since 6⋅4 = 24, (x6)4 = x24

x24

Let's look at one an ext example:

(3x8)4

First, we have the right 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 also x8⋅x8⋅x8⋅x8 = x32, ours answer is:

81x32

Notice this would certainly have likewise been the same as 34⋅x32.

Still confused around multiplying, dividing, or elevating exponents come a power? check out the video clip below to find out a trick for remembering the rules: