Let's go through the questions in the image one by one.

Question 1:

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

Simplification:

• Group the terms involving $\sqrt{2}$ and $\sqrt{3}$: $(2\sqrt{2} - \sqrt{2}) + (-\sqrt{3} + 5\sqrt{3})$ Simplify each part: $(2 - 1)\sqrt{2} = \sqrt{2}$ $(-1 + 5)\sqrt{3} = 4\sqrt{3}$

Thus, the simplified expression is: $\sqrt{2} + 4\sqrt{3}$

The correct answer is option A.

Question 2:

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

Simplification:

• First, group the terms involving $\sqrt{5}$ and $\sqrt{6}$: $(4\sqrt{5} - 9\sqrt{5}) + (5\sqrt{6})$ Simplify the $\sqrt{5}$ terms: $(4 - 9)\sqrt{5} = -5\sqrt{5}$ Also, note that $\sqrt{2} \times 10\sqrt{3}$ simplifies as follows: $\sqrt{2} \times 10\sqrt{3} = 10\sqrt{6}$ So, the expression becomes: $-5\sqrt{5} + 5\sqrt{6} - 10\sqrt{6}$ Simplify the $\sqrt{6}$ terms: $(5 - 10)\sqrt{6} = -5\sqrt{6}$

Thus, the final simplified expression is: $-5\sqrt{5} - 5\sqrt{6}$

The correct answer is option B.

Do you need further details on any step? Here are 5 related questions for additional practice:

1. Simplify $3\sqrt{7} - 2\sqrt{7} + 5\sqrt{3} - 4\sqrt{3}$.
2. Combine like terms in the expression $7\sqrt{6} - 3\sqrt{6} + 4\sqrt{2}$.
3. Simplify $6\sqrt{2} - 4\sqrt{2} + 9\sqrt{5} - 9\sqrt{5}$.
4. How would you simplify $12\sqrt{3} - 7\sqrt{3} + 3\sqrt{2} - 5\sqrt{2}$?
5. What is the result of $\sqrt{8} + 3\sqrt{2} - 2\sqrt{8}$?

Tip: Always group like terms (terms with the same square root) before simplifying to make the process easier.