Sometimes we will have powers that are fractions, such as:

Sometimes we will have powers that are fractions, such as 8^{2/3}. The denominator of the fractions refers to the root and the numerator refers to the power that we raise the root to. We are able to do the root and the power in any order, but it is generally easier do the root first and then the power (especially if we do not have access to a calculator).

**Example 1**

What is:

The denominator of a fractional power is the root. For this question, the denominator is 3. Therefore, we cube root 8. The numerator of the fractional power is the power that we raise the root to. For this example, the numerator is 2, so we square the cube root of 8. We have the calculation below.

Let’s work with the cube root first. We are looking for a number that when multiplied by itself 3 times is 8. This number is 2.

We now square 2, which means that we multiply 2 by itself.

Therefore, 8^{2/3} = 4.

**Example 2**

What is:

The denominator of the fraction is 2, which tells us that we need to square root 16. The numerator of the fractional power is 3, which means that we cube the square root of 16.

Let’s work with the square root of 16 first; we are looking for a number that when multiplied by itself twice is 16. This number is 4.

We now need to cube 4, which means that we multiply 4 by itself 3 times; 4 x 4 x 4.

The answer is 64.

**Example 3**

What is:

The first step to answering this question is to convert the current power from a decimal to a fraction. The current power is 0.5 and this as a fraction is ½.

The denominator of the fraction in the power is 2, which tells us that we need to square root the fraction. The numerator of the fraction is 1, which means that we raise the square root of the fraction to 1.

When we are square rooting this fraction, it is easier to think about this fraction as a numerator and a denominator; find the square root of 25 and 49 separately and then combine the fraction at the end. The square root of 25 is 5 and the square root of 49 is 7. This means that our root is:

We now need to raise this fraction to the power of 1. Anything to the power of 1 is just the base. This means that the answer to this question is ^{5}/_{7}.