The image shows a mathematical expression involving radicals and exponents. It asks to find an equivalent expression for:

$\sqrt{\frac{25x^9y^3}{64x^6y^{11}}}$ assuming $x > 0$ and $y > 0$.

Let's simplify this step by step:

1. Simplify the expression inside the square root: $\frac{25x^9y^3}{64x^6y^{11}} = \frac{25}{64} \cdot \frac{x^9}{x^6} \cdot \frac{y^3}{y^{11}} = \frac{25}{64} \cdot x^{9-6} \cdot y^{3-11} = \frac{25}{64} \cdot x^3 \cdot y^{-8}$

2. Now apply the square root to each term: $\sqrt{\frac{25}{64} \cdot x^3 \cdot y^{-8}} = \frac{\sqrt{25}}{\sqrt{64}} \cdot \sqrt{x^3} \cdot \sqrt{y^{-8}} = \frac{5}{8} \cdot x^{3/2} \cdot y^{-8/2} = \frac{5}{8} \cdot x^{3/2} \cdot y^{-4}$

So the simplified expression is: $\frac{5x^{3/2}}{8y^4}$

Therefore, the correct answer is:

$\boxed{\frac{5x^{\frac{3}{2}}}{8y^4}}$

This corresponds to the last option in the list.

Would you like more details or have any questions about the steps?

Here are 5 related questions:

1. How do you simplify expressions with radicals and exponents?
2. What rules apply when simplifying variables with different powers?
3. How do you handle negative exponents in a fraction?
4. What properties of square roots are essential for simplifying?
5. How can we check the validity of an answer involving radicals and exponents?

Tip: When dealing with exponents, remember to subtract the powers of like bases when dividing.