2-3-2-2-3-2-3-2-2-3-simplify-

Question Number 59764 by ANTARES VY last updated on 14/May/19

\frac{2 + \sqrt{3}}{\sqrt{2} + \sqrt{2 + \sqrt{3}}} + \frac{2 - \sqrt{3}}{\sqrt{2} - \sqrt{2 - \sqrt{3}}} .

simplify .

Answered by Kunal12588 last updated on 14/May/19

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

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

= \sqrt{2} (\frac{6 + \sqrt{3} - 3 + 6 - \sqrt{3} - 3}{9 - 3})

= \sqrt{2}

Commented by Kunal12588 last updated on 14/May/19

f i n d i n g r o o t s o f s u r d s

o f t y p e \sqrt{a \pm \sqrt{b}}

\sqrt{a \pm \sqrt{b}} = \sqrt{m} \pm \sqrt{n}

\Rightarrow a \pm b = (m + n) \pm 2 \sqrt{m n}

\Rightarrow m + n = a, 4 m n = b

\Rightarrow m - n = \sqrt{{(m + n)}^{2} - 4 m n} = \sqrt{a^{2} - b}

\Rightarrow m = \frac{a + \sqrt{a^{2} - b}}{2}, n = \frac{a - \sqrt{a^{2} - b}}{2}

∴ \sqrt{a \pm \sqrt{b}} = \frac{\sqrt{a + \sqrt{a^{2} - b}} \pm \sqrt{a - \sqrt{a^{2} - b}}}{\sqrt{2}}

e x a m p l e

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

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

\sqrt{14 - 3 \sqrt{3}} = \frac{\sqrt{14 + \sqrt{196 - 27}} - \sqrt{14 - \sqrt{196 - 27}}}{\sqrt{2}} = \frac{3 \sqrt{3} - 1}{\sqrt{2}}