Question

1. $-sqrt{18} - sqrt{6} + 2sqrt{2}$
2. $-3sqrt{12} - 2sqrt{27} - 2sqrt{45}$
3. $-sqrt{5} + 3sqrt{5} + 2sqrt{45}$
4. $-2sqrt{54} - 3sqrt{6} + 2sqrt{54}$
5. $3sqrt{8} + 2sqrt{27} + 3sqrt{3}$
6. $3sqrt{54} - 3sqrt{45} + 3sqrt{45}$
7. $2sqrt{12} + 3sqrt{45} + 3sqrt{3}$
8. $-2sqrt{27} - sqrt{54} - sqrt{54}$
9. $4sqrt{72} + 4sqrt{128} - sqrt{96} + 4sqrt{8}$
10. $-3sqrt{5} + 3sqrt{112} + 4sqrt{27} + 2sqrt{45}$
11. $3sqrt{72} - 3sqrt{72} - 2sqrt{6} + 4sqrt{7}$
12. $-3sqrt{7} - 2sqrt{8} - 4sqrt{6} - 2sqrt{8}$

Explanation:

Problem 11: $-\sqrt{18} - \sqrt{6} + 2\sqrt{2}$

Step 1: Simplify $\sqrt{18}$

$\sqrt{18} = \sqrt{9\times2} = 3\sqrt{2}$

Step 2: Substitute and combine like terms

$-\sqrt{18} - \sqrt{6} + 2\sqrt{2} = -3\sqrt{2} - \sqrt{6} + 2\sqrt{2} = (-3\sqrt{2} + 2\sqrt{2}) - \sqrt{6} = -\sqrt{2} - \sqrt{6}$

Step 1: Simplify each square root

$\sqrt{12} = \sqrt{4\times3} = 2\sqrt{3}$, $\sqrt{27} = \sqrt{9\times3} = 3\sqrt{3}$, $\sqrt{45} = \sqrt{9\times5} = 3\sqrt{5}$

Step 2: Substitute and combine like terms

$-3\sqrt{12} - 2\sqrt{27} - 2\sqrt{45} = -3\times2\sqrt{3} - 2\times3\sqrt{3} - 2\times3\sqrt{5} = -6\sqrt{3} - 6\sqrt{3} - 6\sqrt{5} = -12\sqrt{3} - 6\sqrt{5}$

Step 1: Simplify $\sqrt{45}$

$\sqrt{45} = \sqrt{9\times5} = 3\sqrt{5}$

Step 2: Substitute and combine like terms

$-\sqrt{5} + 3\sqrt{5} + 2\sqrt{45} = -\sqrt{5} + 3\sqrt{5} + 2\times3\sqrt{5} = (-\sqrt{5} + 3\sqrt{5}) + 6\sqrt{5} = 2\sqrt{5} + 6\sqrt{5} = 8\sqrt{5}$

Answer:

$-\sqrt{2} - \sqrt{6}$

Problem 12: $-3\sqrt{12} - 2\sqrt{27} - 2\sqrt{45}$