Let's work through each of the five expressions one by one to simplify them.

1. $3(5\sqrt{2})$

This is a straightforward multiplication: $3 \times 5\sqrt{2} = 15\sqrt{2}$ Simplified expression: $15\sqrt{2}$

2. $(2\sqrt{3})(\sqrt{15})$

We can combine the square roots: $2\sqrt{3} \times \sqrt{15} = 2\sqrt{3 \times 15} = 2\sqrt{45}$ Now, simplify $\sqrt{45}$: $\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$ So, the entire expression becomes: $2 \times 3\sqrt{5} = 6\sqrt{5}$ Simplified expression: $6\sqrt{5}$

3. $3(2\sqrt{3} - 4\sqrt{5})$

Distribute the 3: $3 \times 2\sqrt{3} = 6\sqrt{3}, \quad 3 \times -4\sqrt{5} = -12\sqrt{5}$ So, the simplified expression is: $6\sqrt{3} - 12\sqrt{5}$ Simplified expression: $6\sqrt{3} - 12\sqrt{5}$

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

Distribute the $\sqrt{3}$: $\sqrt{3} \times 5\sqrt{6} = 5\sqrt{3 \times 6} = 5\sqrt{18} = 5 \times 3\sqrt{2} = 15\sqrt{2}$ $\sqrt{3} \times 4\sqrt{3} = 4\sqrt{3 \times 3} = 4\sqrt{9} = 4 \times 3 = 12$ So, the expression becomes: $15\sqrt{2} + 12$ Simplified expression: $15\sqrt{2} + 12$

5. $(1 + \sqrt{2})(3 + \sqrt{2})$

Apply the distributive property (FOIL method): $1 \times 3 = 3, \quad 1 \times \sqrt{2} = \sqrt{2}$ $\sqrt{2} \times 3 = 3\sqrt{2}, \quad \sqrt{2} \times \sqrt{2} = 2$ Now, add everything: $3 + \sqrt{2} + 3\sqrt{2} + 2 = 5 + 4\sqrt{2}$ Simplified expression: $5 + 4\sqrt{2}$

If you need more details or further clarifications on any step, feel free to ask!

Relative Questions:

1. How do you simplify the multiplication of two square roots in general?
2. What is the distributive property, and how is it applied to square roots?
3. Why does $\sqrt{2} \times \sqrt{2} = 2$ hold true?
4. How do you simplify complex expressions with multiple square roots?
5. What techniques can be used to simplify $\sqrt{45}$ into $3\sqrt{5}$?

Tip:

When simplifying square roots, always check if the radicand (number inside the square root) can be factored into a perfect square, which allows for further simplification!