Question

a)\(\sqrt{11+6\sqrt{2}}\) -\(\sqrt{2}\)

b)\(\sqrt{28-10\sqrt{3}}\) +5

c)\(\sqrt{4+2\sqrt{3}}\) -\(\sqrt{4-2\sqrt{3}}\)

d)\(3\sqrt{5}\) - \(\sqrt{6-2\sqrt{5}}\)

Answer 1

\(\sqrt{11+6\sqrt{2}}-\sqrt{2}\text{=}\sqrt{9+6\sqrt{2}+4}-\sqrt{2}\)

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

Answer 2

a/ \(\sqrt{11+6\sqrt{2}}-\sqrt{2}=\sqrt{9+6\sqrt{2}+2}-\sqrt{2}=\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{2}=3+\sqrt{2}-\sqrt{2}=3\)

b/ \(\sqrt{28-10\sqrt{3}}+5=\sqrt{25-10\sqrt{3}+3}+5\)

\(=\sqrt{\left(5-\sqrt{3}\right)^2}+5=5-\sqrt{3}+5=25-\sqrt{3}\)

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

\(=\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\)

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

d/ \(3\sqrt{5}-\sqrt{6-2\sqrt{5}}=3\sqrt{5}-\sqrt{5-2\sqrt{5}+1}\)\

\(=3\sqrt{5}-\sqrt{\left(\sqrt{5}-1\right)^2}=3\sqrt{5}-\sqrt{5}+1=2\sqrt{5}+1\)

Answer 3

\(\sqrt{28-10\sqrt{3}}+5=\sqrt{\left(25-10\sqrt{3}+3\right)}+5=\sqrt{\left(5-\sqrt{3}\right)^2}+5=10-\sqrt{3}\)

Answer 4

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

Answer 5

\(3\sqrt{5}-\sqrt{6-2\sqrt{5}}=3\sqrt{5}-\sqrt{\left(1-\sqrt{5}\right)^2}=3\sqrt{5}-\sqrt{5}+1=2\sqrt{5}+1\)