To determine which expression is equivalent to \( x^{\frac{1}{3}} \), let's go through the options given:
A. \( \sqrt{x^3} \):
This expression represents the square root of \( x^3 \), which can be written as \( (x^3)^{1/2} \). Using the exponent rule \( (a^m)^n = a^{m \cdot n} \), this becomes \( x^{3/2} \). Therefore, \( \sqrt{x^3} \) is equivalent to \( x^{3/2} \). This is not the same as \( x^{1/3} \).
B. \( \frac{\pi}{3} \):
This expression is a constant value and does not involve the variable \( x \). Therefore, it cannot be equivalent to \( x^{1/3} \).
C. \( \sqrt[3]{x} \):
This expression represents the cube root of \( x \). The cube root of \( x \) is another way of saying \( x \) raised to the power of \( 1/3 \). Thus, \( \sqrt[3]{x} \) is equivalent to \( x^{1/3} \).
D. \( \frac{1}{x^3} \):
This expression represents the reciprocal of \( x \) raised to the third power. It can be written as \( x^{-3} \). Therefore, \( \frac{1}{x^3} \) is not equivalent to \( x^{1/3} \).
After evaluating all the options, we can conclude that the only expression equivalent to \( x^{\frac{1}{3}} \) is:
C. [tex]\( \sqrt[3]{x} \)[/tex]
