Question

The value of the expression $\frac{1}{\sqrt{11-2\sqrt{30}}}-\frac{3}{\sqrt{7-2\sqrt{10}}}-\frac{4}{\sqrt{8+4\sqrt{3}}}$ after simplification is

• A. $\sqrt{30}$
• B. $2\sqrt{10}$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

$\frac{1}{\sqrt{11-2\sqrt{30}}}-\frac{3}{\sqrt{7-2\sqrt{10}}}-\frac{4}{\sqrt{8+4\sqrt{3}}}$

$={\left(\frac{1}{11-2\sqrt{30}}\right)}^{\frac{1}{2}}-{\left(\frac{3}{7-2\sqrt{10}}\right)}^{\frac{1}{2}}-{\left(\frac{4}{8+4\sqrt{3}}\right)}^{\frac{1}{2}}$

$={\left(\frac{1}{5+6-2\sqrt{5}\sqrt{6}}\right)}^{\frac{1}{2}}-{\left(\frac{3}{5+2-2\sqrt{5}\sqrt{2}}\right)}^{\frac{1}{2}}-{\left(\frac{4}{6+2+2\sqrt{6}\sqrt{2}}\right)}^{\frac{1}{2}}$

$={\left(\frac{1}{(\sqrt{6}-\sqrt{5}{)}^2}\right)}^{\frac{1}{2}}-3{\left(\frac{1}{(\sqrt{5}-\sqrt{2}{)}^2}\right)}^{\frac{1}{2}}-4{\left(\frac{1}{(\sqrt{6}+\sqrt{2}{)}^2}\right)}^{\frac{1}{2}}$

$={\left(\frac{1}{\sqrt{6}-\sqrt{5}}\right)}^{\frac{2}{2}}-3{\left(\frac{1}{\sqrt{5}-\sqrt{2}}\right)}^{\frac{2}{2}}-4{\left(\frac{1}{\sqrt{6}+\sqrt{2}}\right)}^{\frac{2}{2}}$

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

$=(\sqrt{6}+\sqrt{5})-(\sqrt{5}+\sqrt{2})-(\sqrt{6}-\sqrt{2})=0$
Hence, option D is correct.