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 :

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]