To find the expression equivalent to [tex]\( x^{\frac{1}{3}} \)[/tex], let's go through the given options one by one in detail:
1. Option A: [tex]\( \sqrt[3]{x} \)[/tex]
The notation [tex]\( \sqrt[3]{x} \)[/tex] represents the cube root of [tex]\( x \)[/tex]. By definition, taking the cube root of [tex]\( x \)[/tex] is the same as raising [tex]\( x \)[/tex] to the power of [tex]\( \frac{1}{3} \)[/tex]. Therefore,
[tex]\[ \sqrt[3]{x} = x^{\frac{1}{3}} \][/tex]
This matches the given expression exactly.
2. Option B: [tex]\( \frac{1}{x^3} \)[/tex]
The expression [tex]\( \frac{1}{x^3} \)[/tex] represents the reciprocal of [tex]\( x \)[/tex] raised to the power of 3. Simplified in exponential form, it can be written as:
[tex]\[ \frac{1}{x^3} = x^{-3} \][/tex]
This is clearly different from [tex]\( x^{\frac{1}{3}} \)[/tex] since [tex]\( -3 \ne \frac{1}{3} \)[/tex].
3. Option C: [tex]\( \sqrt{x^3} \)[/tex]
The notation [tex]\( \sqrt{x^3} \)[/tex] represents the square root of [tex]\( x^3 \)[/tex]. In exponential form, it can be expressed as:
[tex]\[ \sqrt{x^3} = (x^3)^{\frac{1}{2}} = x^{3 \cdot \frac{1}{2}} = x^{\frac{3}{2}} \][/tex]
This is different from [tex]\( x^{\frac{1}{3}} \)[/tex] because [tex]\( \frac{3}{2} \ne \frac{1}{3} \)[/tex].
4. Option D: [tex]\( \frac{\pi}{3} \)[/tex]
The expression [tex]\( \frac{\pi}{3} \)[/tex] is a numerical value and has no variable component associated with [tex]\( x \)[/tex]. Thus, it cannot be equivalent to [tex]\( x^{\frac{1}{3}} \)[/tex].
Based on this detailed evaluation, the correct equivalent expression to [tex]\( x^{\frac{1}{3}} \)[/tex] is:
Option A: [tex]\( \sqrt[3]{x} \)[/tex]
