https://file.aiquickdraw.com/mathbot/file/172666572581383u5vy81.jpg

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.

Which expression is equivalent to \( \frac{\sqrt{25x^9y^3}}{\sqrt{64x^6y^{11}}} \)? Assume x > 0 and y > 0.

Discover how to simplify expressions involving radicals and exponents, such as \( \frac{\sqrt{25x^9y^3}}{\sqrt{64x^6y^{11}}} \). This guide walks through exponent rules and properties of square roots, making it ideal for students in Grades 9-11.

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