Question

Write with a rational exponent:

(a)$\sqrt[5]{a^2}$

(b)${\left(\sqrt[3]{5ab}\right)}^5$

(c) $\sqrt{{\left(\frac{7xy}{z}\right)}^3}$

Short answer

The rational exponent of $\sqrt[5]{a^2}$is$a^{\frac{2}{5}}$

The rational exponent of ${\left(\sqrt[3]{5ab}\right)}^5$is${\left(5ab\right)}^{\frac{5}{3}}$

The rational exponent of $\sqrt{{\left(\frac{7xy}{z}\right)}^3}$ is${\left(\frac{7xy}{z}\right)}^{\frac{3}{2}}$

Step-by-step solution

Part (a) Step 1. Given

The radical expression$\sqrt[5]{a^2}$

To write the expression with a rational exponent.

Part (a) Step 2. Find numerator and denominator of exponent from the given expression

The numerator of the exponent is exponent, $2$.

The denominator of the exponent is the index of the radical, $5$.

So the rational exponent is ${\left(a\right)}^{\frac{2}{5}}$

Part (b) Step 1. Given

The radical expression ${\left(\sqrt[3]{5ab}\right)}^5$

To write the expression with a rational exponent.

Part (b) Step 2. Find numerator and denominator of exponent from the given expression

The numerator of the exponent is exponent, $5$.

The denominator of the exponent is the index of the radical, $3$.

So the rational exponent is ${\left(5ab\right)}^{\frac{5}{3}}$

Part (c) Step 1. Given

The radical expression$\sqrt{{\left(\frac{7xy}{z}\right)}^3}$

To write the expression with a rational exponent.

Part (c) Step 2. Find numerator and denominator of exponent from the given expression

The numerator of the exponent is exponent, $3$.

The denominator of the exponent is the index of the radical, $2$.

So the rational exponent is ${\left(\frac{7xy}{z}\right)}^{\frac{3}{2}}$