Question

Find the value of $\frac{26-15\sqrt{3}}{[5\sqrt{2}-\sqrt{38+5\sqrt{3}}{]}^2}+\frac{\sqrt{10}+\sqrt{18}}{\sqrt{8}+\sqrt{(3-\sqrt{5})}}.$

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{3}{5}$
• D. $\frac{7}{3}$

Answer 1

The correct option is D $\frac{7}{3}$

$\sqrt{8}+\sqrt{3-\sqrt{5}}$

$=(\sqrt{8}+\sqrt{3-\sqrt{5}})\frac{\sqrt{2}}{\sqrt{2}}$

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

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

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

$\frac{\sqrt{10}+\sqrt{18}}{\sqrt{8}+\sqrt{3-\sqrt{5}}}=$

$\frac{\sqrt{10}+\sqrt{18}}{\frac{1}{\sqrt{2}}(3+\sqrt{5})}=$

$\frac{\sqrt{20}+\sqrt{36}}{3+\sqrt{5}}=\frac{2\sqrt{5}+6}{3+\sqrt{5}}=2$

$5\sqrt{5}-\sqrt{38+5\sqrt{3}}=\frac{1}{\sqrt{2}}\left[\sqrt{2}(5\sqrt{2}-\sqrt{38+5\sqrt{3}})\right]$

$=\frac{1}{\sqrt{2}}[10-\sqrt{76+2\sqrt{75}}]=\frac{1}{\sqrt{2}}[10-\sqrt{{(\sqrt{75}+\sqrt{1})}^2}]$

$=\frac{1}{\sqrt{2}}[10-(\sqrt{75}+1)]=\frac{1}{\sqrt{2}}[9-\sqrt{75}]$

$\frac{26-15\sqrt{3}}{{[5\sqrt{2}-\sqrt{38+5\sqrt{3}}]}^2}=\frac{26-15\sqrt{3}}{{\left[\frac{1}{\sqrt{2}}(9-\sqrt{75})\right]}^2}=\frac{2(26-15\sqrt{3})}{75-2\left(9\right)\$\sqrt{75}}$

$\frac{52-30\sqrt{3}}{156-90\sqrt{3}}$

$=\frac{52-30\sqrt{3}}{3(52-30\sqrt{3})}=\frac{1}{3}$

$\therefore \frac{\sqrt{10+s\sqrt{18}}}{\sqrt{8}+\sqrt{3-\sqrt{5}}}+\frac{26-15\sqrt{3}}{{[5\sqrt{2}-\sqrt{38}+5\sqrt{3}]}^2}=2+\frac{1}{3}=\frac{7}{3}$