# Rational Exponents

## POWERS OF $\frac{1}{2}$

Definition of ${b}^{\frac{1}{2}}$ : (This is read as $b$ to the one-half power.)  If the laws of exponents are to hold, then $\left({b}^{\frac{1}{2}}\right)=$ ${b}^{{\left(\frac{1}{2}\right)}^{2}}=$ ${b}^{1}=b$ . Since the square of ${b}^{\frac{1}{2}}$ is $b$ , ${b}^{\frac{1}{2}}$ is defined to be $\sqrt{b}$ .

Example 1:

Simplify. ${9}^{\frac{1}{2}}$

${9}^{\frac{1}{2}}=\sqrt{9}=3$

In general, raising a number to the $\frac{1}{2}$ power is the same as taking the square root of the number.

## OTHER FRACTIONAL POWERS

Definition of ${b}^{\frac{1}{3}}:{b}^{\frac{1}{3}}$ is defined to be $\sqrt[3]{b}$ , since its cube is $b$ .

Definition of ${b}^{\frac{1}{4}}:{b}^{\frac{1}{4}}$ is defined to be $\sqrt[4]{b}$ , since ${\left({b}^{\frac{1}{4}}\right)}^{4}$ is $b$ .

Definition of ${b}^{\frac{3}{4}}$ :  Using the Power of a Power Law of Exponents in either of two ways:

$\begin{array}{l}{b}^{\frac{3}{4}}={\left({b}^{\frac{1}{4}}\right)}^{3}={\left(\sqrt[4]{b}\right)}^{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\\ {b}^{\frac{3}{4}}={\left({b}^{3}\right)}^{\frac{1}{4}}=\sqrt[4]{{b}^{3}}\end{array}$

Therefore, ${b}^{\frac{3}{4}}$ is defined to be either of the equivalent expressions ${\left(\sqrt[4]{b}\right)}^{3}$ or $\sqrt[4]{{b}^{3}}$ .

The definition of any rational exponent is:

If $p$ and $q$ are integers, $q\ne 0$ and $b$ is a positive real number, then

${b}^{p/q}={\left(\sqrt[q]{b}\right)}^{p}=\sqrt[q]{{b}^{p}}$ .

Example 2:

Simplify. ${27}^{\frac{2}{3}}$

${27}^{\frac{2}{3}}={\left(\sqrt[3]{27}\right)}^{2}={3}^{2}=9$

Notice that in this case computing the root first is easier.  This is usually the case.

## CUBE ROOTS AND OTHER RADICALS

Fractional exponents can also be written as radicals:

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

$\sqrt[5]{x}={x}^{\frac{1}{5}}$

$\sqrt[5]{{x}^{2}}={x}^{\frac{2}{5}}$