To determine which rational exponent represents a cube root, let's follow these steps:
1. Understand Rational Exponents:
- A rational exponent [tex]\( \frac{n}{d} \)[/tex] means that we first take the [tex]\( d \)[/tex]-th root of the base and then raise the result to the power of [tex]\( n \)[/tex].
- Specifically, the [tex]\( d \)[/tex]-th root of a number [tex]\( x \)[/tex] can be written as [tex]\( x^{\frac{1}{d}} \)[/tex].
2. Identify the Cube Root:
- A cube root is the specific case where [tex]\( d = 3 \)[/tex]. Thus, the cube root of [tex]\( x \)[/tex] can be written as [tex]\( x^{\frac{1}{3}} \)[/tex].
3. Match with Given Options:
- Option A: [tex]\( \frac{3}{2} \)[/tex] represents raising to the power of 3 and taking the square root, not a cube root.
- Option B: [tex]\( \frac{1}{3} \)[/tex] specifically represents the cube root.
- Option C: [tex]\( \frac{1}{2} \)[/tex] represents the square root.
- Option D: [tex]\( \frac{1}{4} \)[/tex] represents the fourth root.
4. Conclusion:
- The rational exponent that represents a cube root is [tex]\( \frac{1}{3} \)[/tex].
Therefore, the correct choice is B. [tex]\(\frac{1}{3}\)[/tex].
