# How do you evaluate fractional exponents?

Dec 20, 2014

${x}^{\frac{a}{b}} = \sqrt[b]{{x}^{a}} = {\left(\sqrt[b]{x}\right)}^{a}$

You can just remember this rule, or you can learn about why this is:

fractional exponent $\frac{1}{b}$

So first we're going to look at an expression of the form: ${x}^{\frac{1}{b}}$.
To investigate what this means, we need to go from $x \to {x}^{\frac{1}{b}}$ and then deduce something from it.

${x}^{1} = {x}^{\frac{b}{b}} = {x}^{\frac{1}{b} \cdot b}$
What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself $b$ times.
${x}^{\frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \ldots}$ (b times)

There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get:
${x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \ldots$ (b times)

Now, we still know that this number is equal to $x$. So now we have to think a bit. What number, multiplied by itself b times, gives you $x$.
It's the bth-root of $x$ => ${x}^{\frac{1}{b}} = \sqrt[b]{x}$

For example: ${8}^{\frac{1}{3}}$
If we multiply this by itself 3 times we get:
${8}^{\frac{1}{3}} \cdot {8}^{\frac{1}{3}} \cdot {8}^{\frac{1}{3}} = {8}^{\frac{3}{3}} = 8$
What number multiplied by itself 3 times, gives you 8.
It's of course $\sqrt[3]{8} = 2$

What about $\frac{a}{b}$?
To know what ${x}^{\frac{a}{b}}$ means, we can further rely on our previous findings:
${x}^{\frac{a}{b}} = {x}^{a \cdot \frac{1}{b}} = {x}^{\frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \frac{1}{b} \ldots}$ (a times)
$= {x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \cdot {x}^{\frac{1}{b}} \ldots$ (a times)

Repeated multiplication is equal to exponentiation, so we can write:
$= {\left({x}^{\frac{1}{b}}\right)}^{a} = {\left(\sqrt[b]{x}\right)}^{a}$

You can also bring the exponent in the root:
$= \sqrt[b]{{x}^{a}}$