Question

Rút gọn các biểu thức sau :

a/ \(A=\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}\)

b/ \(\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{6}-\sqrt{2}}\)

c/ \(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}\)

cần gấp !!!! 

Answer 1

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

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

\(=\frac{\sqrt{5}-\sqrt{3}-\sqrt{5}-\sqrt{3}}{5-3}\)

\(=\frac{-2\sqrt{3}}{2}\)

\(=-\sqrt{3}\)

Answer 2

c ) \(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}\)

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

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

\(=\frac{2\sqrt{3}+4}{\sqrt{3}\left(\sqrt{3}+1\right)\left(2+\sqrt{3}\right)}\)

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

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

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

\(=\frac{\sqrt{3}\left(\sqrt{3}-1\right)}{3}\)

\(=\frac{3-\sqrt{3}}{3}\)

\(=1-\frac{\sqrt{3}}{3}\)

Answer 3

b ) \(\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{6}-\sqrt{2}}\)

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

\(=\frac{\sqrt{\left(\sqrt{3}-1\right)^2}}{\sqrt{2}\left(\sqrt{3}-1\right)}\)

\(=\frac{\left|\sqrt{3}-1\right|}{\sqrt{2}\left(\sqrt{3}-1\right)}=\frac{\sqrt{3}-1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\)

\(=\frac{\sqrt{2}}{2}\)

Answer 4

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

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

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

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

\(=\frac{-2\sqrt{3}}{2}\)

\(=-\frac{2\sqrt{3}}{2}\)

\(=-\sqrt{3}\)

b) \(\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{6}-\sqrt{2}}\)

\(=\frac{\sqrt{3}-1}{\sqrt{6}-\sqrt{2}}\)

\(=\frac{\sqrt{2}}{2}\)

c) \(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}\)

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

\(=\frac{3\sqrt{3}+3}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}+\frac{9+5\sqrt{3}}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}-\frac{2\left(2\sqrt{3}+3\right)}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)

\(=\frac{3\sqrt{3}+3+9+5\sqrt{3}-2\left(\sqrt{2}+3\right)}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)

\(=\frac{-2\left(2\sqrt{3}+3\right)+8\sqrt{3}+3+9}{9\sqrt{3}+15}\)

\(=\frac{4\sqrt{3}+6}{9\sqrt{3}+15}\)

\(=\frac{3-\sqrt{3}}{3}\)