Question

match the rational exponent format to its radical expression.
a. $x^{\frac{5}{2}}$
b. $x^{\frac{5}{3}}$
c. $x^{\frac{1}{3}}$
d. $x^{\frac{2}{5}}$

1. $\sqrt{x^5}$
2. $\sqrt5{x}$
3. $\sqrt5{x^2}$
4. $\sqrt3{x^5}$

Explanation:

Step1: Recall rational exponent rule

The rule is $x^{\frac{m}{n}} = \sqrt[n]{x^m}$

Step2: Match a. $x^\frac{5}{2}$

Apply rule: $x^\frac{5}{2} = \sqrt[2]{x^5} = \sqrt{x^5}$, so a → 1

Step3: Match b. $x^\frac{5}{3}$

Apply rule: $x^\frac{5}{3} = \sqrt[3]{x^5}$, so b → 4

Step4: Match c. $x^\frac{1}{3}$

Apply rule: $x^\frac{1}{3} = \sqrt[3]{x^1} = \sqrt[3]{x}$, so c → 2

Step5: Match d. $x^\frac{2}{5}$

Apply rule: $x^\frac{2}{5} = \sqrt[5]{x^2}$, so d → 3

Answer:

a. $x^\frac{5}{2}$ → 1. $\sqrt{x^5}$
b. $x^\frac{5}{3}$ → 4. $\sqrt[3]{x^5}$
c. $x^\frac{1}{3}$ → 2. $\sqrt[3]{x}$
d. $x^\frac{2}{5}$ → 3. $\sqrt[5]{x^2}$