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It seems I couldn't extract any text directly from the image. Let me manually analyze the image for the mathematical content so I can help solve it.The image contains a mathematical problem where you are asked to simplify an expression involving a radical, and all variables are positive. The expression shown is:

\[
\frac{\sqrt{12x^5y^6}}{\sqrt{3x^3y}}
\]

You are then instructed to write the answer in the form \( \frac{A}{B} \) where \( A \) and \( B \) are integers or expressions with no radicals in the denominator.

Let's simplify the expression step by step:

### Step 1: Simplify both radicals separately

- The numerator: \( \sqrt{12x^5y^6} \)
  - \( 12 \) can be broken down into \( 4 \times 3 \), so \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \).
  - \( x^5 \) can be written as \( x^4 \times x = (x^2)^2 \times x \), so \( \sqrt{x^5} = x^2 \sqrt{x} \).
  - \( y^6 = (y^3)^2 \), so \( \sqrt{y^6} = y^3 \).
  
  Therefore, \( \sqrt{12x^5y^6} = 2x^2y^3\sqrt{3x} \).

- The denominator: \( \sqrt{3x^3y} \)
  - \( x^3 \) can be written as \( x^2 \times x \), so \( \sqrt{x^3} = x\sqrt{x} \).
  
  Therefore, \( \sqrt{3x^3y} = \sqrt{3}x\sqrt{x}\sqrt{y} = x\sqrt{3xy} \).

### Step 2: Divide the expressions

Now divide the simplified numerator and denominator:

\[
\frac{2x^2y^3\sqrt{3x}}{x\sqrt{3xy}} = \frac{2x^2y^3}{x} \times \frac{\sqrt{3x}}{\sqrt{3xy}}
\]

First, simplify the rational part:

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

Now simplify the radicals:

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

### Step 3: Combine the results

Now combine everything:

\[
2xy^3 \times \sqrt{\frac{x}{y}} = 2xy^3 \times \frac{\sqrt{x}}{\sqrt{y}} = 2xy^3 \times \frac{1}{y^{1/2}} = 2xy^{5/2}\sqrt{x}
\]

Thus, the simplified expression is:

\[
\boxed{2xy^{5/2}\sqrt{x}}
\]

Let me know if you'd like further clarification or details on this process! 

Here are 5 related questions for further practice:

1. How do you simplify \( \frac{\sqrt{18x^4y^2}}{\sqrt{2x^2y}} \)?
2. What is the process for rationalizing the denominator of \( \frac{3}{\sqrt{5}} \)?
3. How do you simplify \( \sqrt{50x^2} \)?
4. How can you express \( \sqrt{x} \times \sqrt{x} \) in simplified form?
5. What are the general rules for simplifying expressions with radicals?

**Tip**: When simplifying expressions with radicals, always start by breaking down the numbers and variables inside the square root into their prime factors or powers.

Simplify the expression \( \frac{\sqrt{12x^5y^6}}{\sqrt{3x^3y}} \) and express the answer in the form \( \frac{A}{B} \), where \( A \) and \( B \) are expressions with no radicals in the denominator.

Learn how to simplify the radical expression \( \frac{\sqrt{12x^5y^6}}{\sqrt{3x^3y}} \) with a detailed step-by-step solution. This problem covers key algebraic concepts such as radical simplification and properties of radicals. Suitable for students in Grades 9-11.

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