Question

Rút gọn biểu thức:  \(\sqrt{4+2\sqrt{6\sqrt{3}-8}}-\sqrt{5-\sqrt{3}}\)

Answer 1

\(2\sqrt{6\sqrt{3}-8}=2\sqrt{4-\left(3^2-2.3.\sqrt{3}+\left(\sqrt{3}\right)^2\right)}=2\sqrt{2^2-\left(3-\sqrt{3}\right)^2}\)

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

=> \(4+2\sqrt{6\sqrt{3}-8}=\left(\sqrt{3}-1\right)+2\sqrt{\left(\sqrt{3}-1\right).\left(5-\sqrt{3}\right)}+\left(5-\sqrt{3}\right)=\left(\sqrt{\sqrt{3}-1}+\sqrt{5-\sqrt{3}}\right)^2\)=> A = \(\sqrt{4+2\sqrt{6\sqrt{3}-8}}-\sqrt{5-\sqrt{3}}\)

= \(\sqrt{\sqrt{3}-1}+\sqrt{5-\sqrt{3}}\) \(-\sqrt{5-\sqrt{3}}\) = \(\sqrt{\sqrt{3}-1}\)