Prove that

Solution:

(i)

$\frac{1}{3+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+1}$

$=\frac{1}{3+\sqrt{7}} \times \frac{3-\sqrt{7}}{3-\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{5}} \times \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{3}} \times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1}$

$=\frac{3-\sqrt{7}}{(3)^{2}-(\sqrt{7})^{2}}+\frac{\sqrt{7}-\sqrt{5}}{(\sqrt{7})^{2}-(\sqrt{5})^{2}}+\frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5})^{2}-(\sqrt{3})^{2}}+\frac{\sqrt{3}-1}{(\sqrt{3})^{2}-1^{2}}$

$=\frac{3-\sqrt{7}}{9-7}+\frac{\sqrt{7}-\sqrt{5}}{7-5}+\frac{\sqrt{5}-\sqrt{3}}{5-3}+\frac{\sqrt{3}-1}{3-1}$

$=\frac{3-\sqrt{7}}{2}+\frac{\sqrt{7}-\sqrt{5}}{2}+\frac{\sqrt{5}-\sqrt{3}}{2}+\frac{\sqrt{3}-1}{2}$

$=\frac{3-\sqrt{7}+\sqrt{7}-\sqrt{5}+\sqrt{5}-\sqrt{3}+\sqrt{3}-1}{2}$

$=\frac{2}{2}$

$=1$

(ii)

$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}$

$=\frac{1}{1+\sqrt{2}} \times \frac{1-\sqrt{2}}{1-\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}} \times \frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}} \times \frac{\sqrt{4}-\sqrt{5}}{\sqrt{4}-\sqrt{5}}+$ $\frac{1}{\sqrt{5}+\sqrt{6}} \times \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}-\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}} \times \frac{\sqrt{6}-\sqrt{7}}{\sqrt{6}-\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}} \times \frac{\sqrt{7}-\sqrt{8}}{\sqrt{7}-\sqrt{8}}$

$+\frac{1}{\sqrt{8}+\sqrt{9}} \times \frac{\sqrt{8}-\sqrt{9}}{\sqrt{8}-\sqrt{9}}$

$=\frac{1-\sqrt{2}}{1^{2}-(\sqrt{2})^{2}}+\frac{\sqrt{2}-\sqrt{3}}{(\sqrt{2})^{2}-(\sqrt{3})^{2}}+\frac{\sqrt{3}-\sqrt{4}}{(\sqrt{3})^{2}-(\sqrt{4})^{2}}+\frac{\sqrt{4}-\sqrt{5}}{(\sqrt{4})^{2}-(\sqrt{5})^{2}}+\frac{\sqrt{5}-\sqrt{6}}{(\sqrt{5})^{2}-(\sqrt{6})^{2}}+\frac{\sqrt{6}-\sqrt{7}}{(\sqrt{6})^{2}-(\sqrt{7})^{2}}+\frac{\sqrt{7}-\sqrt{8}}{(\sqrt{7})^{2}-(\sqrt{8})^{2}}+\frac{\sqrt{8}-\sqrt{9}}{(\sqrt{8})^{2}-(\sqrt{9})^{2}}$

$=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+\frac{\sqrt{3}-\sqrt{4}}{3-4}+\frac{\sqrt{4}-\sqrt{5}}{4-5}+\frac{\sqrt{5}-\sqrt{6}}{5-6}+\frac{\sqrt{6}-\sqrt{7}}{6-7}+\frac{\sqrt{7}-\sqrt{8}}{7-8}+\frac{\sqrt{8}-\sqrt{9}}{8-9}$

$=\frac{1-\sqrt{2}}{(-1)}+\frac{\sqrt{2}-\sqrt{3}}{(-1)}+\frac{\sqrt{3}-\sqrt{4}}{(-1)}+\frac{\sqrt{4}-\sqrt{5}}{(-1)}+\frac{\sqrt{5}-\sqrt{6}}{(-1)}+\frac{\sqrt{6}-\sqrt{7}}{(-1)}+\frac{\sqrt{7}-\sqrt{8}}{(-1)}+\frac{\sqrt{8}-\sqrt{9}}{(-1)}$

$=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+\sqrt{5}-\sqrt{4}+\sqrt{6}-\sqrt{5}+\sqrt{7}-\sqrt{6}+\sqrt{8}-\sqrt{7}+\sqrt{9}-\sqrt{8}$

$=3-1$

$=2$