The first problem asks us to express the fraction $\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers. The second problem asks us to solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

1. Problem Description

The first problem asks us to express the fraction $\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

The second problem asks us to solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

2. Solution Steps

Problem 1:

First, we rationalize the denominator of the given fraction:

$\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} = \frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} \cdot \frac{2\sqrt{3} + \sqrt{2}}{2\sqrt{3} + \sqrt{2}}$

Now, we multiply the numerators and denominators:

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

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

So, the expression becomes $\frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}} = \frac{\sqrt{6 \cdot 2.5}}{\sqrt{4 \cdot 2.5}} = \frac{\sqrt{15}}{\sqrt{10}}$.

However, we have $\frac{5\sqrt{6}}{10} = \frac{\sqrt{25} \sqrt{6}}{\sqrt{100}} = \frac{\sqrt{150}}{\sqrt{100}}$

Problem 2:

We are given $\sqrt{x+8} + \sqrt{x+1} = 7$.

Isolate one of the square roots: $\sqrt{x+8} = 7 - \sqrt{x+1}$

Square both sides: $(\sqrt{x+8})^2 = (7 - \sqrt{x+1})^2$

$x+8 = 49 - 14\sqrt{x+1} + (x+1)$

$x+8 = 50+x - 14\sqrt{x+1}$

Subtract $x$ from both sides: $8 = 50 - 14\sqrt{x+1}$

$-42 = -14\sqrt{x+1}$

$3 = \sqrt{x+1}$

Square both sides: $9 = x+1$

$x = 8$

Check the solution: $\sqrt{8+8} + \sqrt{8+1} = \sqrt{16} + \sqrt{9} = 4 + 3 = 7$. Thus, $x=8$ is a solution.

3. Final Answer

For the first problem, the answer is $\frac{\sqrt{150}}{\sqrt{100}}$.

For the second problem, the answer is 8.