Fractional exponents are exponents that can be written as fractions. But first, a review of radicals.

 

In the problem , the answer is the number that when multiplied by itself will get 9. The answer is 3. Now think about it this way, “What number multiplied twice gives you 9?” The answer is still 3. Notice that it is not “What number multiplied by 2?”, but rather “what number multiplied twice.” Notice the difference? It is very important to understand it this way in order to study radicals any further.

Now take the previous problem,, to be entirely correct, it needs to be written as . Both radicals are saying the same things only written different. Therefore,

 

=

Use this knowledge even further.

 

? Notice the 3 in front, instead of the 2 commonly used. This is asking for the cubed root of z. Or, in other words, what number times itself and then times itself again will get z? Basically what number, x, where x * x * x = z, will give this result?

 

The way to get this is just like on square roots, the perfect squares are 1,4,9,25…, know the perfect cubes, 1,8,27,…, to answer this. 2 * 2 * 2 =8, so the cubed root of 8 is 2.

Here is a small chart of some of the cubes

 

1³=1

 

2³=8

 

3³=27

 

4³= 64

 

5³= 125

 

To find more of these cubes, use a calculator. On most calculators there is a key that says x^y. First enter the base, 2, push the x^y key, then push 3, and last push =. The result should be 8.

Example

 

 

Answer! By knowing that 125 is 5³or 5(5)(5)= 125.

 

 

 

 

Example

 

 

First break it down into perfect cubes. 128= 64*2.

 

 

 

.

4

 

Answer! It will not break down any further.

 

 

 

When looking at cubed roots, notice that a lot of the same properties hold that were true for square roots, only there is one major exception.

 

 

Example

 

= -1

 

Answer!

 

 

 

 

Why is it not imaginary like in square roots?

By definition of cubed roots, a number is being multiplied by itself three times. Since 3 is an odd number one will end up with a negative because (-1)(-1)(-1)= -1

 

Any time there is an even numbered root of a number, one cannot use negatives without using imaginary #’s. But anytime there is an odd numbered root, negatives will work as in the previous example.

 

 

 

Example

 

= -4

 

Answer! (-4)(-4)(-4) = -64

 

Following is the definition of fractional (sometimes called rational) exponents.

 

=

 

or

 

=

 

Look at a calculator and find the key, or the key. This is the button used for problems like this.

 

Example

 

 

Understand the answer is whatever number multiplies by itself 5 times and gets 16807, so rewrite it using fractional exponents.

 

 

Put 16807 in a calculator and press the key. Next, press 5 and then press =. The result should be 7. If not, keep on trying using a different order.

 

7

 

Answer! 7 * 7 * 7 *7 * 7 = 16807.

 

 

 

Expanding on the previous definition there is a new rule.

 

=

 

or

 

=

 

What the example is saying is “c to the 3/2 power”. In solving it, raise c to the third power (c * c * c) first, and then take the square root of it (that is where the 2 comes in). Or, take the square root first, then cube the result of that. It doesn’t matter which order is used. Use whichever one is easiest for the problem. Following are a couple of examples, one of each kind.

 

Example

 

 

It will be easiest to raise 2 to the third power first. 2³ = 8.

 

8 can be written as 4 * 2.

 

= 2.

 

 

Answer!

 

 

 

 

Example

 

On this one, the square root of 4 is 2. So do that first and raise it to the fifth power later.

2⁵

 

2 * 2 * 2 * 2 * 2 = 32.

32

 

Answer!

 

 

 

 

 

Example

 

 

Before using a calculator realize that .

5²

25

 

Answer!

 

 

 

Always simplify anytime during an expression or equation, it will make it easier later.