Let's go over the first question:

The expression given is:

$2\sqrt{2} - \sqrt{3} + 5\sqrt{3} - \sqrt{2}$

We simplify this by combining like terms:

• $2\sqrt{2} - \sqrt{2} = \sqrt{2}$
• $-\sqrt{3} + 5\sqrt{3} = 4\sqrt{3}$

So, the simplified form of the expression is:

$\sqrt{2} + 4\sqrt{3}$

This matches option a. So, the correct answer is a.

For the second question:

The expression is:

$4\sqrt{5} + 5\sqrt{6} - 9\sqrt{5} - \sqrt{2} \times 10\sqrt{3}$

1. Combine the terms with $\sqrt{5}$: $4\sqrt{5} - 9\sqrt{5} = -5\sqrt{5}$

2. The $5\sqrt{6}$ stays as it is.

3. Simplify $\sqrt{2} \times 10\sqrt{3}$: $\sqrt{2} \times 10\sqrt{3} = 10\sqrt{6}$

So, the expression becomes:

$-5\sqrt{5} + 5\sqrt{6} - 10\sqrt{6}$

Finally, combine the $\sqrt{6}$ terms:

$5\sqrt{6} - 10\sqrt{6} = -5\sqrt{6}$

Thus, the simplified form is:

$-5\sqrt{5} - 5\sqrt{6}$

This matches option a.

Would you like further details or have any questions about these solutions?

Here are 5 related questions for more practice:

1. Simplify $3\sqrt{7} - 2\sqrt{7} + 5\sqrt{7}$.
2. What is $\sqrt{8} + \sqrt{18}$ in its simplest form?
3. Simplify $4\sqrt{3} - 7\sqrt{3} + 6\sqrt{3}$.
4. Find the value of $\sqrt{50} + \sqrt{2}$.
5. Simplify $2\sqrt{12} - 3\sqrt{3} + 4\sqrt{27}$.

Tip: When simplifying square roots, always group terms with the same radicals first before combining them.