We are asked to express $\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

1. Problem Description

We are asked to express $\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

2. Solution Steps

We need to rationalize the denominator of the given expression. Multiply both numerator and denominator by the conjugate of the denominator, which is $2\sqrt{3}+\sqrt{2}$:

$\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}} = \frac{(3\sqrt{2}-\sqrt{3})(2\sqrt{3}+\sqrt{2})}{(2\sqrt{3}-\sqrt{2})(2\sqrt{3}+\sqrt{2})}$

Expanding the numerator:

$(3\sqrt{2}-\sqrt{3})(2\sqrt{3}+\sqrt{2}) = 3\sqrt{2} \cdot 2\sqrt{3} + 3\sqrt{2} \cdot \sqrt{2} - \sqrt{3} \cdot 2\sqrt{3} - \sqrt{3} \cdot \sqrt{2} = 6\sqrt{6} + 6 - 6 - \sqrt{6} = 5\sqrt{6}$

Expanding the denominator:

$(2\sqrt{3}-\sqrt{2})(2\sqrt{3}+\sqrt{2}) = (2\sqrt{3})^2 - (\sqrt{2})^2 = 4 \cdot 3 - 2 = 12 - 2 = 10$

So the expression becomes:

$\frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}}$

Multiply the numerator and denominator by $\sqrt{\frac{5}{5}} = 1$:

$\frac{5\sqrt{6}}{10} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{5 \sqrt{60}}{10}$.

This step does not help.

Rewriting the expression as $\frac{\sqrt{6}}{2}$:

We want to express this in the form $\frac{\sqrt{m}}{\sqrt{n}}$. We can write $2$ as $\sqrt{4}$.

So, $\frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}}$. This does not match the provided options.

We want to match one of the options. Let us square $\frac{5\sqrt{6}}{\sqrt{10}}$:

$\left( \frac{5\sqrt{6}}{\sqrt{10}}\right)^2 = \frac{25 \cdot 6}{10} = \frac{150}{10} = 15$.

Square root to get $\sqrt{15}$

Let us square the given expression:

$\left(\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}}\right)^2 = \frac{18 - 6\sqrt{6} + 3}{12 - 4\sqrt{6} + 2} = \frac{21 - 6\sqrt{6}}{14 - 4\sqrt{6}}$.

This is not equal to

1

5.

The expression simplifies to $\frac{\sqrt{6}}{2}$, so we want to find which of the answer options is equal to this.

(a) $\frac{\sqrt{6}}{\sqrt{10}} = \frac{\sqrt{6}}{\sqrt{10}}$

(b) $\frac{\sqrt{150}}{\sqrt{100}} = \frac{\sqrt{25 \cdot 6}}{\sqrt{100}} = \frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2}$

(c) $\frac{2\sqrt{6}}{\sqrt{10}} = \frac{2\sqrt{6}}{\sqrt{10}}$

(d) $\frac{5\sqrt{6}}{\sqrt{10}} = \frac{5\sqrt{6}}{\sqrt{10}}$

3. Final Answer

(b) $\frac{\sqrt{150}}{\sqrt{100}}$