Question

Write with a rational exponent:

(a)$\sqrt{x^5}$

(b)${\left(\sqrt[4]{3y}\right)}^3$

(c) $\sqrt{{\left(\frac{2m}{3n}\right)}^5}$

Short answer

The rational exponent of $\sqrt{x^5}$is$x^{\frac{5}{2}}$

The rational exponent of ${\left(\sqrt[4]{3y}\right)}^3$is${\left(3y\right)}^{\frac{3}{4}}$

The rational exponent of$\sqrt{{\left(\frac{2m}{3n}\right)}^5}$ is ${\left(\frac{2m}{3n}\right)}^{\frac{5}{2}}$

Step-by-step solution

Part (a) Step 1. Given

The radical expression $\sqrt{x^5}$

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, $5$.

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

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

Part (b) Step 1. Given

The radical expression ${\left(\sqrt[4]{3y}\right)}^3$

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, $3$.

The denominator of the exponent is the index of the radical, $4$.

So the rational exponent is ${\left(3y\right)}^{\frac{3}{4}}$

Part (c) Step 1. Given

The radical expression$\sqrt{{\left(\frac{2m}{3n}\right)}^5}$

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, $5$.

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

So the rational exponent is ${\left(\frac{2m}{3n}\right)}^{\frac{5}{2}}$