Question

rut gon bieu thuc a) \(\sqrt{15+2\sqrt{5}-\sqrt[]{21-4\sqrt{5}}}\)

b)\(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)

Answer 1

a) \(\sqrt{15+2\sqrt{5}-\sqrt{21-4\sqrt{5}}}\)

\(=\sqrt{15+2\sqrt{5}-\sqrt{\left(1-2\sqrt{5}\right)^2}}\)

\(=\sqrt{15+2\sqrt{5}-\left(2\sqrt{5}-1\right)}\)

\(=\sqrt{15+2\sqrt{5}-\left(2\sqrt{5}-1\right)}\)

\(=\sqrt{15+2\sqrt{5}-2\sqrt{5}+1}\)

\(=\sqrt{16}\)

\(=4\)

b) \(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)

\(=\sqrt[4]{5-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)

\(=\sqrt[4]{5-\sqrt{3-\sqrt{\left(3-2\sqrt{5}\right)^2}}}\)

\(=\sqrt[4]{5-\sqrt{3-\left(2\sqrt{5}-3\right)}}\)

\(=\sqrt[4]{5-\sqrt{3-2\sqrt{5}+3}}\)

\(=\sqrt[4]{5-\sqrt{6-2\sqrt{5}}}\)

\(=\sqrt[4]{5-\sqrt{\left(1-\sqrt{5}\right)^2}}\)

\(=\sqrt[4]{5-\left(\sqrt{5}-1\right)}\)

\(=\sqrt[4]{5-\sqrt{5}+1}\)

\(=\sqrt[4]{6-\sqrt{5}}\)