Question

In the following exercises, write with a rational exponent.

$ⓐ\sqrt[4]{5x}ⓑ\sqrt[8]{9y}ⓒ7\sqrt[5]{3z}$

Short answer

Part (a). $\sqrt[4]{5x}={\left(5x\right)}^{\frac{1}{4}}$

Part (b). $\sqrt[8]{9y}={\left(9y\right)}^{\frac{1}{8}}$

Part (c).$7\sqrt[5]{3z}=7{\left(3z\right)}^{\frac{1}{5}}$

Step-by-step solution

Part (a) Step 1. Rational exponent of  5x4

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="1644750007013" $\sqrt[4]{5x}={\left(5x\right)}^{\frac{1}{4}}$

Part (b) Step 1. Rational exponent of  9y8

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

The index is$8$

The denominator of the exponent will be $8$.

role="math" localid="1644750026382" $\sqrt[8]{9y}={\left(9y\right)}^{\frac{1}{8}}$

Part (c) Step 1. Rational exponent of  73z5

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

The index is $5$

The denominator of the exponent will be $5$

role="math" localid="1644750055676" $7\sqrt[5]{3z}=7{\left(3z\right)}^{\frac{1}{5}}$