Question

select the correct answer. which expression is equivalent to $x^{\frac{1}{3}}$? a. $\sqrt3{x}$ b. $\sqrt{x^3}$ c. $\frac{x}{3}$ d. $\frac{1}{x^3}$

Explanation:

Step1: Recall radical-exponent rule

For any real number $x$ and positive integers $m,n$ ($n>1$), $x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$, and $x^{\frac{1}{n}} = \sqrt[n]{x}$.

Step2: Match with given expression

We have $x^{\frac{1}{3}}$, so applying the rule, this equals $\sqrt[3]{x}$.

Step3: Evaluate other options

• Option B: $\sqrt{x^3}=x^{\frac{3}{2}}$, not equivalent.
• Option C: $\frac{x}{3}$ is a linear term, not equivalent.
• Option D: $\frac{1}{x^3}=x^{-3}$, not equivalent.

Answer:

A. $\sqrt[3]{x}$