Question

\(a,\sqrt{\left(\sqrt{5}-3\right)^2}+\sqrt{9-4\sqrt{5}}\)

\(b,\left(\sqrt{10}+\sqrt{2}\right)\left(6-2\sqrt{5}\right)\sqrt{3+\sqrt{5}}\)

\(c,\frac{\sqrt{7}-5}{2}-\frac{6}{\sqrt{7}-2}+\frac{1}{3+\sqrt{7}}+\frac{2}{5+2\sqrt{7}}\)

Answer 1

a) \(=\left|\sqrt{5}-3\right|+\sqrt{5-2\sqrt{20}+4}\)

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

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

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

\(=1\)

 

 

Answer 2

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

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

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

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

\(=16\)

 

Answer 3

c)\(=\frac{\sqrt{7}-5}{2}-\frac{6\left(\sqrt{7}+2\right)}{7-4}+\frac{\left(3-\sqrt{7}\right)}{9-7}+\frac{2\left(5-2\sqrt{7}\right)}{25-28}\)

\(=\frac{\sqrt{7}-5}{2}-\frac{6\left(\sqrt{7}+2\right)}{3}+\frac{3-\sqrt{7}}{2}-\frac{2\left(5-2\sqrt{7}\right)}{3}\)

\(=\frac{\sqrt{7}-\sqrt{7}-5+3}{2}-\frac{6\sqrt{7}+12+10-4\sqrt{7}}{3}\)

\(=-1-\frac{2\sqrt{7}+22}{3}\)

\(=\frac{-3-2\sqrt{7}-22}{3}\)

\(=-\frac{25+2\sqrt{7}}{3}\)