Question

Tính giá trị ccacs biểu thức:
\(\frac{2}{\sqrt{6}-2}+\frac{2}{\sqrt{6}+2}+\frac{5}{\sqrt{6}}\)

 \(\frac{1}{\sqrt{3}+\sqrt{2}-\sqrt{5}}-\frac{1}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)

Answer 1

\(\frac{2}{\sqrt{6}-2}+\frac{2}{\sqrt{6}+2}+\frac{5}{\sqrt{6}}.\)

\(=\frac{\sqrt{2}.\sqrt{2}}{\sqrt{2}\left(\sqrt{3}-1\right)}+\frac{\sqrt{2}.\sqrt{2}}{\sqrt{2}\left(\sqrt{3}+1\right)}+\frac{5}{\sqrt{6}}\)

\(=\frac{\sqrt{2}\left(\sqrt{3}+1\right)}{3-1}+\frac{\sqrt{2}\left(\sqrt{3}-1\right)}{3+1}+\frac{5}{\sqrt{6}}\)

\(=\frac{\left(\sqrt{3}+1\right)}{\sqrt{2}}+\frac{\sqrt{3}-1}{\sqrt{8}}+\frac{5}{\sqrt{6}}\)

\(=...\)

Answer 2

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

\(=\frac{2.\left(\sqrt{6}+2+\sqrt{6}-2\right)}{\left(\sqrt{6}-2\right)\left(\sqrt{6}+2\right)}+\frac{5\sqrt{6}}{6}\)

\(=\frac{4\sqrt{6}}{6-2^2}+\frac{5\sqrt{6}}{6}=2\sqrt{6}+\frac{5\sqrt{6}}{6}\)

\(=\frac{17\sqrt{6}}{6}\)

\(b,\frac{1}{\sqrt{3}+\sqrt{2}-\sqrt{5}}-\frac{1}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)

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

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

\(=\frac{2\sqrt{5}}{5+2\sqrt{6}-5}=\sqrt{\frac{5}{6}}\)