Question

Rút gọn:

a) \(\sqrt{14-6\sqrt{5}}\) +\(\sqrt{6+2\sqrt{5}}\)

b) \(\dfrac{\sqrt{10}+10}{1+\sqrt{10}}-\dfrac{5\sqrt{2}-2\sqrt{5}}{\sqrt{5}-\sqrt{2}}\)

Answer 1

Lời giải:

a) Ta có:

\(14-6\sqrt{5}=14-2\sqrt{45}=9+5-2\sqrt{9.5}=(\sqrt{9}-\sqrt{5})^2=(3-\sqrt{5})^2\)

\(\Rightarrow \sqrt{14-6\sqrt{5}}=3-\sqrt{5}\)

\(6+2\sqrt{5}=5+1+2\sqrt{5.1}=(\sqrt{5}+1)^2\)

\(\Rightarrow \sqrt{6+2\sqrt{5}}=\sqrt{5}+1\)

Do đó: \(\sqrt{14-6\sqrt{5}}+\sqrt{6+2\sqrt{5}}=3-\sqrt{5}+\sqrt{5}+1=4\)

b)

\(\frac{\sqrt{10}+10}{1+\sqrt{10}}-\frac{5\sqrt{2}-2\sqrt{5}}{\sqrt{5}-\sqrt{2}}=\frac{\sqrt{10}(1+\sqrt{10})}{1+\sqrt{10}}-\frac{\sqrt{10}(\sqrt{5}-\sqrt{2})}{\sqrt{5}-\sqrt{2}}\)

\(=\sqrt{10}-\sqrt{10}=0\)