﻿ Formula and examples of how to simplify fraction exponents

# Simplify Fraction Exponents

Formula and examples of how to simplify Fraction exponents

#### Examples of Rewriting Fractional Exponents

If you've ever calculated the square root of a number then you were actually using a fraction exponent! The square root of a number is the same as raising that number to an exponent of the fraction $$\frac 1 2$$

$\sqrt 3 = 3 ^ {\red { \frac 1 2} }$

$\sqrt[3] 8 = 8 ^ {\red { \frac 1 3} }$

$\sqrt[4] 81 = 81 ^ {\red { \frac 1 4} }$

#### So how does this work?

We can use one of the laws of exponents to explain how fractional exponents work.

As you probably already know $$\sqrt{9} \cdot \sqrt{9} = 9$$ . Well, let's look at how that would work with rational (read: fraction ) exponents . Since we now know $$\sqrt{9} = 9^{\frac 1 2 }$$ . We can express $$\sqrt{9} \cdot \sqrt{9} = 9$$ as :

$\\ 9^{\frac 1 2 } \cdot 9^{\frac 1 2 } = 9^{\frac 1 2 + \frac 1 2 } \\ = \boxed{ 9 ^1 }$

We can do the same thing with $$\sqrt[3] 8 \cdot \sqrt[3] 8 \cdot \sqrt[3] 8 = 8$$

$\\ 8^{\frac 1 3} \cdot 8^{\frac 1 3 } \cdot 8^{\frac 1 3 } = 8^{\frac 1 3 + \frac 1 3+ \frac 1 3 } \\ = \boxed{ 8 ^1 }$

### General Formula

#### With fractional exponents whose numerator is 1

Below is the general formula for a fractional exponent with a numerator of 1.

$\sqrt[n] x = x ^ {\frac 1 n}$

$$\frac 1 n$$ is another way of asking: What number can you multiply by itself n times to get x?

#### When the numerator is not 1

Below is a specific example illustrating the formula for fraction exponents when the numerator is not one. There are two ways to simplify a fraction exponent such $$\frac 2 3$$ . You can either apply the numerator first or the denominator. See the example below.

Example