8.3 Simplify Rational Exponents

We want to write each expression in the form $\sqrt[n]{a}.$

ⓐ
$\begin{array}{cccc}& & & \hfill x^{\frac{1}{2}}\hfill \\ \begin{array}{c}\text{The denominator of the rational exponent is 2, so}\hfill \\ \text{the index of the radical is 2. We do not show the}\hfill \\ \text{index when it is 2.}\hfill \end{array}\hfill & & & \hfill \sqrt{x}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill y^{\frac{1}{3}}\hfill \\ \begin{array}{c}\text{The denominator of the exponent is 3, so the}\hfill \\ \text{index is 3.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{y}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill z^{\frac{1}{4}}\hfill \\ \begin{array}{c}\text{The denominator of the exponent is 4, so the}\hfill \\ \text{index is 4.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{z}\hfill \end{array}$

We want to write each radical in the form $a^{\frac{1}{n}}.$

ⓐ
$\begin{array}{cccc}& & & \hfill \sqrt{5y}\hfill \\ \begin{array}{c}\text{No index is shown, so it is 2.}\hfill \\ \text{The denominator of the exponent will be 2.}\hfill \end{array}\hfill & & & \hfill {\left(5y\right)}^{\frac{1}{2}}\hfill \\ \begin{array}{c}\text{Put parentheses around the entire}\hfill \\ \text{expression}5y.\hfill \end{array}\hfill & & & \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \sqrt[3]{4x}\hfill \\ \begin{array}{c}\text{The index is 3, so the denominator of the}\hfill \\ \text{exponent is 3. Include parentheses}\left(4x\right).\hfill \end{array}\hfill & & & \hfill \begin{array}{}\\ \\ \hfill {\left(4x\right)}^{\frac{1}{3}}\hfill \end{array}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill 3\sqrt[4]{5z}\hfill \\ \begin{array}{c}\text{The index is 4, so the denominator of the}\hfill \\ \text{exponent is 4. Put parentheses only around}\hfill \\ \text{the}5z\text{since 3 is not under the radical sign.}\hfill \end{array}\hfill & & & \hfill \begin{array}{}\\ \\ \\ \\ \hfill 3{\left(5z\right)}^{\frac{1}{4}}\hfill \end{array}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill {25}^{\frac{1}{2}}\hfill \\ \text{Rewrite as a square root.}\hfill & & & \hfill \sqrt{25}\hfill \\ \text{Simplify.}\hfill & & & \hfill 5\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill {64}^{\frac{1}{3}}\hfill \\ \text{Rewrite as a cube root.}\hfill & & & \hfill \sqrt[3]{64}\hfill \\ \text{Recognize 64 is a perfect cube.}\hfill & & & \hfill \sqrt[3]{4^3}\hfill \\ \text{Simplify.}\hfill & & & \hfill 4\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill {256}^{\frac{1}{4}}\hfill \\ \text{Rewrite as a fourth root.}\hfill & & & \hfill \sqrt[4]{256}\hfill \\ \text{Recognize 256 is a perfect fourth power.}\hfill & & & \hfill \sqrt[4]{4^4}\hfill \\ \text{Simplify.}\hfill & & & \hfill 4\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill {\left(\mathrm{-16}\right)}^{\frac{1}{4}}\hfill \\ \text{Rewrite as a fourth root.}\hfill & & & \hfill \sqrt[4]{\mathrm{-16}}\hfill \\ & & & \hfill \sqrt[4]{{\left(\mathrm{-2}\right)}^4}\hfill \\ \text{Simplify.}\hfill & & & \hfill \text{No real solution.}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \text{−}{16}^{\frac{1}{4}}\hfill \\ \begin{array}{c}\text{The exponent only applies to the 16.}\hfill \\ \text{Rewrite as a fouth root.}\hfill \end{array}\hfill & & & \hfill \text{−}\sqrt[4]{16}\hfill \\ \text{Rewrite 16 as}{2}^4.\hfill & & & \hfill \text{−}\sqrt[4]{2^4}\hfill \\ \text{Simplify.}\hfill & & & \hfill \mathrm{-2}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill {\left(16\right)}^{-\frac{1}{4}}\hfill \\ \text{Rewrite using the property}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \hfill \frac{1}{{\left(16\right)}^{\frac{1}{4}}}\hfill \\ \text{Rewrite as a fourth root.}\hfill & & & \hfill \frac{1}{\sqrt[4]{16}}\hfill \\ \text{Rewrite 16 as}{2}^4.\hfill & & & \hfill \frac{1}{\sqrt[4]{2^4}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \frac{1}{2}\hfill \end{array}$

We want to use $a^{\frac{m}{n}}=\sqrt[n]{a^m}$ to write each radical in the form $a^{\frac{m}{n}}.$

ⓐ

ⓑ

ⓒ

We will rewrite the expression as a radical first using the defintion, $a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m.$ This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

ⓐ
$\begin{array}{cccc}& & & \hfill {125}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{The power of the radical is the numerator of the exponent, 2.}\hfill \\ \text{The index of the radical is the denominator of the}\hfill \\ \text{exponent, 3.}\hfill \end{array}\hfill & & & \hfill {\left(\sqrt[3]{125}\right)}^2\hfill \\ \text{Simplify.}\hfill & & & \hfill {\left(5\right)}^2\hfill \\ & & & \hfill 25\hfill \end{array}$

ⓑ We will rewrite each expression first using $a^{\text{−}n}=\frac{1}{a^n}$ and then change to radical form.
$\begin{array}{cccc}& & & \hfill {16}^{-\frac{3}{2}}\hfill \\ \\ \\ \text{Rewrite using}{a}^{\text{−}n}=\frac{1}{a^n}\hfill & & & \hfill \frac{1}{{16}^{\frac{3}{2}}}\hfill \\ \\ \\ \begin{array}{c}\text{Change to radical form. The power of the radical is the}\hfill \\ \text{numerator of the exponent, 3. The index is the denominator}\hfill \\ \text{of the exponent, 2.}\hfill \end{array}\hfill & & & \hfill \frac{1}{{\left(\sqrt{16}\right)}^3}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{1}{4^3}\hfill \\ \\ \\ & & & \hfill \frac{1}{64}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill {32}^{-\frac{2}{5}}\hfill \\ \\ \\ \text{Rewrite using}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \hfill \frac{1}{{32}^{\frac{2}{5}}}\hfill \\ \\ \\ \text{Change to radical form.}\hfill & & & \hfill \frac{1}{{\left(\sqrt[5]{32}\right)}^2}\hfill \\ \\ \\ \text{Rewrite the radicand as a power.}\hfill & & & \hfill \frac{1}{{\left(\sqrt[5]{2^5}\right)}^2}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{1}{2^2}\hfill \\ \\ \\ & & & \hfill \frac{1}{4}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \text{−}{25}^{\frac{3}{2}}\hfill \\ \text{Rewrite in radical form.}\hfill & & & \hfill \text{−}{\left(\sqrt{25}\right)}^3\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill \text{−}{\left(5\right)}^3\hfill \\ \text{Simplify.}\hfill & & & \hfill \mathrm{-125}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \text{−}{25}^{-\frac{3}{2}}\hfill \\ \text{Rewrite using}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \hfill \text{−}\left(\frac{1}{{25}^{\frac{3}{2}}}\right)\hfill \\ \text{Rewrite in radical form.}\hfill & & & \hfill \text{−}\left(\frac{1}{{\left(\sqrt{25}\right)}^3}\right)\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill \text{−}\left(\frac{1}{{\left(5\right)}^3}\right)\hfill \\ \text{Simplify.}\hfill & & & \hfill -\frac{1}{125}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill {\left(\mathrm{-25}\right)}^{\frac{3}{2}}\hfill \\ \text{Rewrite in radical form.}\hfill & & & \hfill {\left(\sqrt{\mathrm{-25}}\right)}^3\hfill \\ \begin{array}{c}\text{There is no real number whose square root}\hfill \\ \text{is}\mathrm{-25}.\hfill \end{array}\hfill & & & \hfill \text{Not a real number.}\hfill \end{array}$

ⓐ The Product Property tells us that when we multiply the same base, we add the exponents.

$\begin{array}{cccc}& & & \hfill x^{\frac{1}{2}}·x^{\frac{5}{6}}\hfill \\ \\ \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & \hfill x^{\frac{1}{2}+\frac{5}{6}}\hfill \\ \\ \\ \text{Add the fractions.}\hfill & & & \hfill x^{\frac{8}{6}}\hfill \\ \\ \\ \text{Simplify the exponent.}\hfill & & & \hfill x^{\frac{4}{3}}\hfill \end{array}$

ⓑ The Power Property tells us that when we raise a power to a power, we multiply the exponents.

$\begin{array}{cccc}& & & \hfill {\left(z^9\right)}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \hfill z^{9·\frac{2}{3}}\hfill \\ \text{Simplify.}\hfill & & & \hfill z^6\hfill \end{array}$

ⓒ The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

$\begin{array}{cccc}& & & \hfill \frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\hfill \\ \\ \\ & & & \hfill \frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\hfill \\ \\ \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \hfill \frac{1}{x^{\frac{5}{3}-\frac{1}{3}}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{1}{x^{\frac{4}{3}}}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill {\left(27u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & \hfill {\left(27\right)}^{\frac{2}{3}}{\left(u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \text{Rewrite 27 as a power of 3.}\hfill & & & \hfill {\left(3^3\right)}^{\frac{2}{3}}{\left(u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \hfill \left(3^2\right)\left(u^{\frac{1}{3}}\right)\hfill \\ \text{Simplify.}\hfill & & & \hfill 9u^{\frac{1}{3}}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill {\left(m^{\frac{2}{3}}{n}^{\frac{1}{2}}\right)}^{\frac{3}{2}}\hfill \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & \hfill {\left(m^{\frac{2}{3}}\right)}^{\frac{3}{2}}{\left(n^{\frac{1}{2}}\right)}^{\frac{3}{2}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \hfill m{n}^{\frac{3}{4}}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \frac{x^{\frac{3}{4}}·x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Product Property in the numerator,}\hfill \\ \text{add the exponents.}\hfill \end{array}\hfill & & & \hfill \frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property, subtract the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & \hfill x^{\frac{8}{4}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill x^2\hfill \end{array}$