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.