To determine which expression is equivalent to [tex]\(\sqrt{\frac{25 x^9 y^3}{64 x^6 y^{11}}}\)[/tex], let's break down the given expression step by step.
### Step 1: Simplify the Fraction Inside the Square Root
First, we simplify the fraction under the square root:
[tex]\[
\frac{25 x^9 y^3}{64 x^6 y^{11}}
\][/tex]
We can simplify the fraction by dividing both the numerator and the denominator by the common factors. Let's simplify the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[
\frac{x^9}{x^6} = x^{9-6} = x^3
\][/tex]
[tex]\[
\frac{y^3}{y^{11}} = y^{3-11} = y^{-8} = \frac{1}{y^8}
\][/tex]
So the fraction simplifies to:
[tex]\[
\frac{25 x^3}{64 y^8}
\][/tex]
### Step 2: Rewrite the Simplified Fraction
Now the fraction inside the square root is:
[tex]\[
\frac{25 x^3}{64 y^8}
\][/tex]
### Step 3: Apply the Square Root
Next, we apply the square root to the entire fraction:
[tex]\[
\sqrt{\frac{25 x^3}{64 y^8}}
\][/tex]
We can split the square root across the numerator and the denominator:
[tex]\[
\frac{\sqrt{25 x^3}}{\sqrt{64 y^8}}
\][/tex]
### Step 4: Simplify the Square Roots Individually
Now, let's simplify each part separately:
[tex]\[
\sqrt{25 x^3} = \sqrt{25} \cdot \sqrt{x^3} = 5 \cdot \sqrt{x^3} = 5 \cdot x^{3/2} = 5 \cdot x^{1.5}
\][/tex]
[tex]\[
\sqrt{64 y^8} = \sqrt{64} \cdot \sqrt{y^8} = 8 \cdot y^{8/2} = 8 \cdot y^4
\][/tex]
So, the expression becomes:
[tex]\[
\frac{5 \cdot x^{1.5}}{8 \cdot y^4} = \frac{5 \sqrt{x^3}}{8 y^4} = \frac{5 (x \cdot \sqrt{x})}{8 y^4} = \frac{5 \sqrt{x}}{8 y^4}
\][/tex]
### Step 5: Compare to Provided Options
Out of the given options, the expression equivalent to [tex]\(\sqrt{\frac{25 x^9 y^3}{64 x^6 y^{11}}}\)[/tex] is:
[tex]\[
\frac{5 \sqrt{x}}{8 y^4}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{5 \sqrt{x}}{8 y^4}}
\][/tex]
