What space exponents?

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

You are watching: The number or expression in a power that is multiplied by itself

Exponents are useful because they let us write lengthy numbers in a reduce form. Because that instance, this number is really large:


But you could write that this method as an exponent:


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


It also could be written as one exponent:


Scientists regularly use index number to convey very large numbers and also very small ones. You'll watch them often in algebra problems too.

Understanding exponents

As you experienced in the video, exponents are written like this: 43 (you'd read it as 4 come the 3rd power). All exponents have actually two parts: the base, which is the number being multiplied; and also the power, i beg your pardon is the number of times you main point the base.


Because our base is 4 and our power is 3, we’ll need to multiply 4 by chin three times.

43 = 4 ⋅ 4 ⋅ 4 = 64

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

Occasionally, you could see the exact same exponent written prefer this: 5^3. Don’t worry, it’s specifically the very same number—the base is the number come the left, and also the strength is the number come the right. Relying on the type of calculator girlfriend use—and particularly if you’re utilizing the calculator on your phone or computer—you may need come input the exponent this way to calculation it.

Exponents come the first and 0th power

How would certainly you leveling these exponents?

71 70

Don’t feel negative if you’re confused. Also if you feeling comfortable with other exponents, it’s not evident how to calculation ones through powers of 1 and 0. Luckily, these exponents follow an easy rules:

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

Operations through exponents

How would you fix this problem?

22 ⋅ 23

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

How about this one?

x3 / x2

Or this one?

2x2 + 2x2

While you can’t precisely solve these difficulties without more information, you deserve to simplify them. In algebra, you will often be asked to carry out calculations on exponents through variables together the base. Fortunately, it’s straightforward to add, subtract, multiply, and also divide this exponents.

Adding exponents

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

x2 + x2 = 2x2

How about 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 could have noticed that we only looked at difficulties where the exponents us were adding had the very same variable and also power. This is because you can only include exponents if your bases and exponents room exactly the same. So you can include these below because both terms have actually the very same variable (r) and the same power (7):

4r7 + 9r7

You can never add any of these as they’re written. This expression has variables through 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 adding them. Because that example, deserve to you figure out how to simplify this expression?

5x2 - 4x2

5-4 is 1, for this reason if you stated 1x2, or simply x2, you’re right. Remember, similar to with adding exponents, you have the right to only subtract exponents v the same power and base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying index number is simple, yet the means you do it might surprise you. To main point exponents, add the powers. Because that instance, take it this expression:

x3 ⋅ x4

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

x3 ⋅ x4 = x7

What about this expression?

3x2 ⋅ 2x6

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

3x2 ⋅ 2x6 = 6x8

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

Dividing exponents

Dividing exponents is similar to multiply them. Rather of adding the powers, girlfriend subtract them. Take it this expression:

x8 / x2

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

x8 / x2 = x6

What around this one?

10x4 / 2x2

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

Raising a strength to a power

Sometimes you could see an equation prefer this:


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


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

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

(x5)3 = x15

Did you an alert that 5⋅3 also equals 15? Remember, multiplication is the very same as adding something more than once. That way we have the right to think of 5+5+5, i beg your pardon is what us did earlier, as 5 time 3. Therefore, when you advanced a power come a power you deserve to multiply the exponents.

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Let's look in ~ one an ext example:


Since 6⋅4 = 24, (x6)4 = x24


Let's look in ~ one much more example:


First, we can rewrite this as:


Remember in multiplication, stimulate does no matter. Therefore, we deserve to rewrite this again as:


Since 3⋅3⋅3⋅3 = 81 and also x8⋅x8⋅x8⋅x8 = x32, our answer is:


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

Still confused about multiplying, dividing, or elevating exponents come a power? examine out the video below to discover a trick because that remembering the rules: