The problem asks to determine which expression is equivalent to $\sqrt[3]{15^5}$.

Solution:

1. Recall that the cube root of a number $a^b$ can be expressed as $a^{b/3}$. So, $\sqrt[3]{15^5} = 15^{5/3}$.

2. Comparing the given options:

   • $15^2$ corresponds to $15^2$, which is not the same as $15^{5/3}$.
   • $15^8$ corresponds to $15^8$, which is also not equivalent.
   • $15^{3/5}$ represents $15^{3/5}$, not $15^{5/3}$.
   • $15^{5/3}$ matches exactly.

Correct Answer:

The correct answer is $15^{5/3}$.

Would you like a breakdown of exponent rules or further clarification?

Here are 5 related questions to expand your understanding:

1. How would you simplify $\sqrt[4]{16^6}$?
2. What is the difference between $\sqrt[3]{x^2}$ and $x^{2/3}$?
3. If $x^{a/b}$ is given, how can it be rewritten in terms of roots and powers?
4. What happens if the numerator of the fractional exponent is greater than the denominator?
5. Why does $a^{m/n} = \sqrt[n]{a^m}$ hold for fractional exponents?

Tip: Fractional exponents provide a convenient way to write roots and powers together, which simplifies many mathematical operations!