Question

In the following exercises, write with a rational exponent.

$ⓐ\sqrt{21p}ⓑ\sqrt[4]{48q}ⓒ4\sqrt[6]{36r}$

Short answer

Part (a). $\sqrt{21p}={\left(21\right)}^{\frac{1}{2}}$

Part (b). $\sqrt[4]{8q}={\left(8q\right)}^{\frac{1}{4}}$

Part (c). $4\sqrt[6]{36r}=4{\left(36r\right)}^{\frac{1}{6}}$

Step-by-step solution

Part (a) Step 1. Rational exponent of 21p 

We know that, $\sqrt[n]{a}=a^{\frac{1}{n}}$

The index is$2$

The denominator of the exponent will be $2$.

role="math" localid="1644757867840" $\sqrt{21p}={\left(21\right)}^{\frac{1}{2}}$

Part (b) Step 1. Rational exponent of  8q4

We know that,$\sqrt[n]{a}=a^{\frac{1}{n}}$

The index is$4$

The denominator of the exponent will be $4$.

role="math" localid="1644757850228" $\sqrt[4]{8q}={\left(8q\right)}^{\frac{1}{4}}$

Part (c) Step 1. Rational exponent of  436r6

We know that,$\sqrt[n]{a}=a^{\frac{1}{n}}$

The index is$6$

The denominator of the exponent will be$6$

role="math" localid="1644757919525" $4\sqrt[6]{36r}=4{\left(36r\right)}^{\frac{1}{6}}$