Simplify

Solution:

(i)

$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$

$=\frac{4+\sqrt{5}}{4-\sqrt{5}} \times \frac{4+\sqrt{5}}{4+\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}} \times \frac{4-\sqrt{5}}{4-\sqrt{5}}$

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

$=\frac{16+5+8 \sqrt{5}+16+5-8 \sqrt{5}}{16-5}$

$=\frac{42}{11}$

(ii)

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

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

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

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

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

$=\sqrt{3}-\sqrt{2}-\sqrt{5}-\sqrt{3}+\sqrt{2}+\sqrt{5}$

$=0$

(iii)

$\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$

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

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

$=\frac{4+3+4 \sqrt{3}}{4-3}+\frac{4+3-4 \sqrt{3}}{4-3}+\frac{3+1-2 \sqrt{3}}{3-1}$

$=7+4 \sqrt{3}+7-4 \sqrt{3}+\frac{4-2 \sqrt{3}}{2}$

$=14+2-\sqrt{3}$

$=16-\sqrt{3}$

(iv)

$\frac{2 \sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6 \sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8 \sqrt{3}}{\sqrt{6}+\sqrt{2}}$

$=\frac{2 \sqrt{6}}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}+\frac{6 \sqrt{2}}{\sqrt{6}+\sqrt{3}} \times \frac{\sqrt{6}-\sqrt{3}}{\sqrt{6}-\sqrt{3}}-\frac{8 \sqrt{3}}{\sqrt{6}+\sqrt{2}} \times \frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}-\sqrt{2}}$

$=\frac{2 \sqrt{6} \times \sqrt{3}-2 \sqrt{6} \times \sqrt{2}}{(\sqrt{3})^{2}-(\sqrt{2})^{2}}+\frac{6 \sqrt{2} \times \sqrt{6}-6 \sqrt{2} \times \sqrt{3}}{(\sqrt{6})^{2}-(\sqrt{3})^{2}}-\frac{8 \sqrt{3} \times \sqrt{6}-8 \sqrt{3} \times \sqrt{2}}{(\sqrt{6})^{2}-(\sqrt{2})^{2}}$

$=\frac{2 \sqrt{18}-2 \sqrt{12}}{3-2}+\frac{6 \sqrt{12}-6 \sqrt{6}}{6-3}-\frac{8 \sqrt{18}-8 \sqrt{6}}{6-2}$

$=2 \sqrt{18}-2 \sqrt{12}+\frac{6 \sqrt{12}-6 \sqrt{6}}{3}-\frac{8 \sqrt{18}-8 \sqrt{6}}{4}$

$=2 \sqrt{18}-2 \sqrt{12}+2 \sqrt{12}-2 \sqrt{6}-2 \sqrt{18}+2 \sqrt{6}$

$=0$