Dealing with fractional exponents
How fractional exponents are related to roots
In this lesson we’ll work with both positive and negative fractional exponents. Remember that when ???a??? is a positive real number, both of these equations are true:
???x^{-a}=\frac{1}{x^a}???
???\frac{1}{x^{-a}} = x^a???
The rule for fractional exponents:
When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable example ???x^{\frac{a}{b}}???, where ???a??? and ???b??? are positive real numbers and ???x??? is a real number, ???a??? is the power and ???b??? is the root.
???x^{\frac{a}{b}}??? ???=??? ???\sqrt[b]{x^a}???
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Changing fractional exponents into roots and vice versa
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An example where we change the fractional exponent into a root in order to simplify the expression
Example
Simplify the expression.
???4^{\frac{3}{2}}???
In the fractional exponent, ???3??? is the power and ???2??? is the root, which means we can rewrite the expression as
???\sqrt{4^3}???
???\sqrt{4 \cdot 4 \cdot 4}???
???\sqrt{64}???
???8???
in a fractional exponent, think of the numerator as an exponent, and the denominator as the root
Another rule for fractional exponents:
To make a problem easier to solve you can break up the exponents by rewriting them. For example, you can write ???x^{\frac{a}{b}}??? as
???\left[(x)^a\right]^{\frac{1}{b}}???
or as
???\left[(x)^{\frac{1}{b}}\right]^a???
Let’s do a few examples.
Let's do another example.
Example
Simplify the expression.
???\left(\frac{1}{9}\right)^{\frac{3}{2}}???
???9??? is a perfect square so it can simplify the problem to find the square root first. We can rewrite the expression by breaking up the exponent.
???\left[\left(\frac{1}{9}\right)^{\frac{1}{2}}\right]^3???
Raising a value to the power ???1/2??? is the same as taking the square root of that value, so we get
???\left[\sqrt{\frac{1}{9}}\right]^3???
???\left(\frac{\sqrt{1}}{\sqrt{9}}\right)^3???
???\left(\frac{1}{3}\right)^3???
This is the same as
???\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)???
???\frac{1}{27}???
How to get rid of fractional exponents
Example
Write the expression without fractional exponents.
???\left(\frac{1}{6}\right)^{\frac{3}{2}}???
We can rewrite the expression by breaking up the exponent.
???\left[\left(\frac{1}{6}\right)^3\right]^{\frac{1}{2}}???
???\left(\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}\right)^{\frac{1}{2}}???
Raising a value to the power ???1/2??? is the same as taking the square root of that value, so we get
???\sqrt{\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}}???
???\sqrt{\frac{1}{216}}???
???\frac{\sqrt{1}}{\sqrt{216}}???
???\frac{1}{\sqrt{36 \cdot 6}}???
???\frac{1}{\sqrt{36} \sqrt{6}}???
???\frac{1}{6\sqrt{6}}???
We need to rationalize the denominator.
???\frac{1}{6\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}???
???\frac{\sqrt{6}}{6 \cdot 6}???
???\frac{\sqrt{6}}{36}???
What happens if you have a negative fractional exponent?
You should deal with the negative sign first, then use the rule for the fractional exponent.
Example
Write the expression without fractional exponents.
???4^{-\frac{2}{5}}???
First, we’ll deal with the negative exponent. Remember that when ???a??? is a positive real number, both of these equations are true:
???x^{-a}=\frac{1}{x^a}???
???\frac{1}{x^{-a}} = x^a???
Therefore,
???4^{-\frac{2}{5}}???
???\frac{1}{4^{\frac{2}{5}}}???
In the fractional exponent, ???2??? is the power and ???5??? is the root, which means we can rewrite the expression as
???\frac{1}{\sqrt[5]{4^2}}???
???\frac{1}{\sqrt[5]{16}}???
