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]
