Formula and examples of how to simplify fraction exponents

Simplify Rational Expressions worksheet
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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.