Let's analyze and simplify the expression from the image. The given expression is:

$\left(\frac{\sqrt[3]{125}}{512}\right)^{\frac{1}{3}}$

To simplify, let's proceed step by step:

1. Evaluate $\sqrt[3]{125}$:

   • Since $125 = 5^3$, we have $\sqrt[3]{125} = 5$.

   So the expression simplifies to:

   $\left(\frac{5}{512}\right)^{\frac{1}{3}}$

2. Evaluate $(\frac{5}{512})^{\frac{1}{3}}$:

   • We can write this as: $\frac{\sqrt[3]{5}}{\sqrt[3]{512}}$.
   • Since $512 = 2^9$, we have $\sqrt[3]{512} = 2^3 = 8$.

   Therefore, the expression becomes:

   $\frac{\sqrt[3]{5}}{8}$

Thus, the simplified form of the expression is:

$\boxed{\frac{\sqrt[3]{5}}{8}}$

Would you like further clarification or more details on any step?

Here are 5 related questions to expand your understanding:

1. What is the general rule for simplifying fractional exponents?
2. How do you evaluate cube roots for larger numbers?
3. Can this simplification method be applied to fourth or fifth roots?
4. What is the difference between a cube root and a square root?
5. How would this problem change if the exponent was $\frac{2}{3}$ instead of $\frac{1}{3}$?

Tip: When simplifying roots, always try to express numbers as powers of smaller base numbers to make it easier to evaluate!