Question

Rút gọn các biểu thức :

\(a,\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)

\(b,\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\sqrt{4-\sqrt{15}}\)

Answer 1

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

b/ \(=\frac{\left(8+2\sqrt{15}\right)}{2}.\left(\sqrt{5}-\sqrt{3}\right)\sqrt{8-2\sqrt{15}}\)

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

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

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