Question

Tính giá trị biểu thức: `A=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}` `B=\sqrt{2+2\sqrt6+2\sqrt3+2\sqrt2}-\sqrt{5+2\sqrt6}`

Answer 1

\(A=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}\)

\(=\dfrac{1}{\sqrt{2}}\cdot\left(\sqrt{8+2\sqrt{15}}+\sqrt{8-2\sqrt{15}}-2\sqrt{6-2\sqrt{5}}\right)\)

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

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

\(=\dfrac{1}{\sqrt{2}}\cdot2=\sqrt{2}\)