9.8 Rational exponents

Solution

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

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

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

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

Solution

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

ⓐ
$\begin{array}{cccc}& & & \sqrt{x}\hfill \\ \begin{array}{c}\text{No index is shown, so it is 2.}\hfill \\ \text{The denominator of the exponent will be 2.}\hfill \end{array}\hfill & & & x^{\frac{1}{2}}\hfill \end{array}$

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

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

Solution

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

1. ⓐ
   $\begin{array}{cccc}& & & \sqrt{5y}\hfill \\ \begin{array}{c}\text{No index is shown, so it is 2.}\hfill \\ \text{The denominator of the exponent will be 2.}\hfill \end{array}\hfill & & & {\left(5y\right)}^{\frac{1}{2}}\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & \sqrt[3]{4x}\hfill \\ \begin{array}{c}\text{The index is 3, so the denominator of the}\hfill \\ \text{exponent is 3.}\hfill \end{array}\hfill & & & {\left(4x\right)}^{\frac{1}{3}}\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccc}& & & 3\sqrt[4]{5z}\hfill \\ \begin{array}{c}\text{The index is 4, so the denominator of the}\hfill \\ \text{exponent is 4.}\hfill \end{array}\hfill & & & 3{\left(5z\right)}^{\frac{1}{4}}\hfill \end{array}$

Solution

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

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

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