Question

Answer 1

1.

\(\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)

2.

\(\frac{2}{2+\sqrt{3}}=\frac{2(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}=\frac{4-2\sqrt{3}}{4-3}=4-2\sqrt{3}\)

3.

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

4.

\(\frac{1}{2\sqrt{13}}=\frac{\sqrt{13}}{2.13}=\frac{\sqrt{13}}{26}\)

5.

\(\frac{10}{9\sqrt{31}}=\frac{10\sqrt{31}}{9.31}=\frac{10\sqrt{31}}{279}\)

 

Answer 2

6.

\(\frac{1}{2+\sqrt{5}}+\frac{1}{2-\sqrt{5}}=\frac{2-\sqrt{5}+2+\sqrt{5}}{(2-\sqrt{5})(2+\sqrt{5})}=\frac{4}{4-5}=-4\)

7.

\(\frac{2}{\sqrt{7}}+\frac{4}{\sqrt{11}}=\frac{2\sqrt{7}}{7}+\frac{4\sqrt{11}}{11}=\frac{28\sqrt{11}+22\sqrt{7}}{77}\)

8.

\(\frac{6\sqrt{5}+5\sqrt{6}}{\sqrt{30}}=\frac{\sqrt{30}(\sqrt{6}+\sqrt{5})}{\sqrt{30}}=\sqrt{6}+\sqrt{5}\)

9.

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

10.

\(\frac{\sqrt{22}-\sqrt{33}}{2-\sqrt{6}}=\frac{\sqrt{11}(\sqrt{2}-\sqrt{3})}{\sqrt{2}(\sqrt{2}-\sqrt{3})}=\frac{\sqrt{11}}{\sqrt{2}}=\frac{\sqrt{22}}{2}\)