# What is a rational exponent?

Jun 11, 2015

A rational exponent is an exponent of the form $\frac{m}{n}$ for two integers $m$ and $n$, with the restriction $n \ne 0$.

${x}^{\frac{m}{n}}$ is basically the same as $\sqrt[n]{{x}^{m}}$

#### Explanation:

Some general rules for exponents are:

${x}^{0} = 1$

${x}^{1} = x$

${x}^{-} 1 = \frac{1}{x}$

${x}^{a} \cdot {x}^{b} = {x}^{a + b}$

${\left({x}^{a}\right)}^{b} = {x}^{a \cdot b}$

If $n$ is a positive integer then

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

From these rules, we can deduce:

${\left(\sqrt[n]{x}\right)}^{m} = {\left({x}^{\frac{1}{n}}\right)}^{m} = {x}^{\frac{1}{n} \cdot m}$

$= {x}^{\frac{m}{n}}$

$= {x}^{m \cdot \frac{1}{n}} = {\left({x}^{m}\right)}^{\frac{1}{n}} = \sqrt[n]{{x}^{m}}$