https://file.aiquickdraw.com/mathbot/file/1726391344432l115z4jp.jpg

The expression in the image is:

\[
\frac{\sqrt{x^3 y}}{\sqrt{x y^3}}
\]

Let's simplify this step by step.

### Step 1: Combine the radicals
Both the numerator and the denominator are square roots. We can combine the entire expression under one square root:

\[
\frac{\sqrt{x^3 y}}{\sqrt{x y^3}} = \sqrt{\frac{x^3 y}{x y^3}}
\]

### Step 2: Simplify the fraction inside the square root
Now, simplify the fraction:

\[
\frac{x^3 y}{x y^3} = \frac{x^{3-1} y^{1-3}} = x^2 y^{-2}
\]

### Step 3: Apply the square root
Now apply the square root to each part:

\[
\sqrt{x^2 y^{-2}} = \sqrt{x^2} \cdot \sqrt{y^{-2}} = x \cdot y^{-1}
\]

Since \( y^{-1} = \frac{1}{y} \), the final result is:

\[
\frac{x}{y}
\]

Thus, the simplified expression is:

\[
\frac{x}{y}
\]

Would you like more details, or do you have any other questions?

Here are 5 related questions to explore further:
1. How would the result change if the exponents of \(x\) and \(y\) were different?
2. What happens if you have cube roots instead of square roots?
3. How do you simplify similar expressions involving higher powers?
4. What is the general rule for dividing radicals?
5. How does the presence of negative exponents affect simplification?

**Tip**: When simplifying fractions under square roots, always look for ways to cancel terms by subtracting exponents.

Simplify the expression: \( \frac{\sqrt{x^3 y}}{\sqrt{x y^3}} \)

Learn how to simplify the radical expression \( \frac{\sqrt{x^3 y}}{\sqrt{x y^3}} \). This step-by-step guide covers the application of exponents, radical simplifications, and algebraic rules. Suitable for Grades 9-10.

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