Question

\(\left(2+\sqrt{2}\right)\sqrt{3-\sqrt{8}}\)

Answer 1

\(\left(2+\sqrt{2}\right)\sqrt{3-\sqrt{8}}\)

=\(2\sqrt{3-\sqrt{8}}+\sqrt{6-2\sqrt{8}}\)=\(\sqrt{12-4\sqrt{8}}+2-\sqrt{2}\)

=\(\sqrt{8}-2+2-\sqrt{2}=\sqrt{8}-\sqrt{2}=\sqrt{2}\)

Answer 2

\(\left(2+\sqrt{2}\right)\sqrt{3-\sqrt{8}}\)

\(=\sqrt{2}\left(\sqrt{2}+1\right)\sqrt{3-\sqrt{8}}\)

\(=\sqrt{6-2\sqrt{8}}\left(\sqrt{2}+1\right)\)

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

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

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

\(=\sqrt{2}\)