Let's break down the questions and solve them one by one:

1. Solve the square root expressions:

a. $(2 \sqrt{3} + \sqrt{2})(2 \sqrt{3} - \sqrt{2})$

This is a difference of squares:

$(a + b)(a - b) = a^2 - b^2$

Here:

• $a = 2 \sqrt{3}$
• $b = \sqrt{2}$

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

Answer: 10

b. $\sqrt{3}(5 \sqrt{6} + 4 \sqrt{5})$

Distribute $\sqrt{3}$:

$\sqrt{3} \times 5 \sqrt{6} + \sqrt{3} \times 4 \sqrt{5}$

First term: $\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3 \sqrt{2}, \quad 5 \times 3 \sqrt{2} = 15 \sqrt{2}$

Second term: $\sqrt{3} \times \sqrt{5} = \sqrt{15}, \quad 4 \times \sqrt{15} = 4 \sqrt{15}$

So, the result is:

$15 \sqrt{2} + 4 \sqrt{15}$

Answer: $15 \sqrt{2} + 4 \sqrt{15}$

2. Rationalize the following:

a. $\frac{2 \sqrt{10}}{\sqrt{7} - \sqrt{5}}$

To rationalize this expression, multiply both the numerator and the denominator by the conjugate of the denominator $\sqrt{7} + \sqrt{5}$:

$\frac{2 \sqrt{10}}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}$

This gives:

$\frac{2 \sqrt{10}(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}$

First, calculate the denominator:

$(\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2$

Now, the numerator:

$2 \sqrt{10} \times \sqrt{7} = 2 \sqrt{70}, \quad 2 \sqrt{10} \times \sqrt{5} = 2 \sqrt{50} = 10 \sqrt{2}$

So the full expression becomes:

$\frac{2 \sqrt{70} + 10 \sqrt{2}}{2}$

This simplifies to:

$\sqrt{70} + 5 \sqrt{2}$

Answer: $\sqrt{70} + 5 \sqrt{2}$

b. $\frac{\sqrt{5}}{\sqrt{8}}$

Simplify the square roots:

$\frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2 \sqrt{2}}$

To rationalize, multiply both the numerator and the denominator by $\sqrt{2}$:

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

Answer: $\frac{\sqrt{10}}{4}$

Would you like further clarifications or details on any of these steps?

Here are 5 related questions you can try:

1. Simplify $(\sqrt{7} + 2)(\sqrt{7} - 2)$.
2. Rationalize $\frac{1}{\sqrt{5} + 2}$.
3. Multiply $3 \sqrt{2} \times 4 \sqrt{3}$.
4. Find the simplified form of $\frac{\sqrt{12}}{\sqrt{3}}$.
5. Simplify $4 \sqrt{6} - 2 \sqrt{6}$.

Tip: When rationalizing denominators, always look for the conjugate to simplify the expression efficiently!