Question

1.\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)

2.\(\sqrt{9-4\sqrt{5}}-\sqrt{9+\sqrt{80}}\)

3.\(\sqrt{17-12\sqrt{2}}-\sqrt{24-8\sqrt{8}}\)

4.\(\sqrt{3+2\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)

5.\(\sqrt{8+2\sqrt{15}}-\sqrt{8-2\sqrt{15}}\)

6.\(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)

7.\(\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}\)

Answer 1

https://i.imgur.com/g7mbF2P.jpg

Answer 2

1. \(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)

\(=\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)

\(=\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{2}\)

\(=2\sqrt{2}\)

Answer 3

2.\(\sqrt{9-4\sqrt{5}}-\sqrt{9+\sqrt{80}}\)

\(=\sqrt{9-4\sqrt{5}}-\sqrt{9+4\sqrt{5}}\)

\(=\sqrt{\left(\sqrt{5}-2\right)^2}-\sqrt{\left(\sqrt{5}+2\right)^2}\)

\(=\sqrt{5}-2-\sqrt{5}-2=-4\)

Answer 4

3.\(\sqrt{17-12\sqrt{2}}-\sqrt{24-8\sqrt{8}}\)

\(=\sqrt{\left(3-2\sqrt{2}\right)^2}-\sqrt{\left(2\sqrt{2}-4\right)^2}\)

\(=3-2\sqrt{2}+2\sqrt{2}-4=-1\)

Answer 5

4.\(\sqrt{3+2\sqrt{2}}-\sqrt{6-4\sqrt{2}}\)

\(=\sqrt{\left(\sqrt{2}+1\right)^2}-\sqrt{\left(2-\sqrt{2}\right)^2}\)

\(=\sqrt{2}+1-2+\sqrt{2}\)

\(=-1+2\sqrt{2}\)

Answer 6

5.\(\sqrt{8+2\sqrt{15}}-\sqrt{8-2\sqrt{15}}\)

\(=\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)

\(=\sqrt{5}+\sqrt{3}-\sqrt{5}+\sqrt{3}\)

\(=2\sqrt{3}\)

Answer 7

6.\(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)

\(=\sqrt{\left(2\sqrt{2}-3\right)^2}+\sqrt{\left(2\sqrt{2}+3\right)^2}\)

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

Answer 8

7.\(\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}\)

\(=\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\)

\(=\sqrt{5}+1+\sqrt{5}-1\)

\(=2\sqrt{5}\)