Will you square 4

Algebra Level 3

$\large \sqrt5 \div \frac { \sqrt { 3-\sqrt { 5 } } }{ \sqrt { 2 } +\sqrt { 7-3\sqrt { 5 } } } = \ ?$

The answer is 5.

2 solutions

Chew-Seong Cheong
Jul 23, 2015

$\begin{aligned} \sqrt{5} \div \frac{\sqrt{3-\sqrt{5}}} {\sqrt{2} + \sqrt{7-3\sqrt{5}}} & = \frac {\sqrt{5}\left( \sqrt{2} + \sqrt{7-3\sqrt{5}} \right)} {\sqrt{3-\sqrt{5}}} \\ & = \frac {\sqrt{5}\left( \sqrt{2} + \sqrt{\dfrac{14-6\sqrt{5}}{2}} \right)} {\sqrt{\dfrac{6-2\sqrt{5}}{2}}} \\ & = \frac {\sqrt{5}\left( \dfrac{2 + \sqrt{(3 - \sqrt{5})^2}}{\sqrt{2}} \right)} {\sqrt{\dfrac{(\sqrt{5} - 1)^2}{2}}} \\ & = \frac {\sqrt{5}\left(2 + 3 - \sqrt{5} \right)} {\sqrt{5}-1} \\ & = \frac {\sqrt{5} \left(5 - \sqrt{5} \right) \left(\sqrt{5} + 1 \right)} {4} \\ & = \frac {\sqrt{5} \dot{} \sqrt{5} \left(\sqrt{5} - 1 \right) \left(1 + \sqrt{5} \right)} {4} \\ & = \frac {5(4)} {4} = \boxed{5} \end{aligned}$

Thanks Sir. 2nd line was a very smart move...

Asif Rochy - 5 years, 10 months ago

Thanks. You have to learn to read the mind of the problem writer. There must be certain tricks needed to solve the problem easily.

Chew-Seong Cheong - 5 years, 10 months ago

Shashank Ojha
Jul 25, 2015

Will you square 4

The answer is 5.

2 solutions

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