Select the correct answer.

Which expression is equivalent to [tex]$x^{\frac{1}{3}}$[/tex]?

A. [tex]$\sqrt{x^3}$[/tex]
B. [tex]$\frac{\pi}{3}$[/tex]
C. [tex]$\sqrt[3]{x}$[/tex]
D. [tex]$\frac{1}{x^3}$[/tex]



Answer :

Let's understand the expression given in the question, \( x^{\frac{1}{3}} \).

The exponent \(\frac{1}{3}\) indicates the cube root of \(x\). In mathematical terms, \( x^{\frac{1}{3}} \) means the same as saying "the number which, when raised to the power of 3, gives \(x\)."

We have four options to evaluate:

1. Option A: \( \sqrt{x^3} \)

This expression represents the square root of \( x^3 \). It is not equivalent to \( x^{\frac{1}{3}} \) because taking the square root of \( x^3 \) and taking the cube root of \( x \) are different operations.

2. Option B: \( \frac{\pi}{3} \)

This expression represents a fraction involving the constant \(\pi\). It does not relate to taking powers or roots of the variable \( x \), so it is not equivalent to \( x^{\frac{1}{3}} \).

3. Option C: \( \sqrt[3]{x} \)

This expression represents the cube root of \( x \). By definition, \( x^{\frac{1}{3}} \) means the cube root of \( x \), so this option is indeed equivalent to \( x^{\frac{1}{3}} \).

4. Option D: \( \frac{1}{x^3} \)

This expression represents the reciprocal of \( x^3 \). This is not equivalent to \( x^{\frac{1}{3}} \) since taking the reciprocal of \( x \) raised to the power of 3 is different from taking the cube root of \( x \).

After going through each option, we see that the correct answer is:

C. [tex]\( \sqrt[3]{x} \)[/tex]