Confusion with a particular surd simplification

**Code2004** · January 20th 2013, 08:55 PM

Hey guys, I'm stuck bad on this simplification:

$\sqrt{201-\sqrt{401}} = \frac{\sqrt{401}-1}{\sqrt{2}}$

I'd highly appreciate a swift response.

Thanks guys!

**Prove It** · January 20th 2013, 09:11 PM

Originally Posted by Code2004

Hey guys, I'm stuck bad on this simplification:

$\sqrt{201-\sqrt{401}} = \frac{\sqrt{401}-1}{\sqrt{2}}$

I'd highly appreciate a swift response.

Thanks guys!

$\displaystyle \begin{align*} x &= \frac{\sqrt{401} - 1}{\sqrt{2}} \\ \sqrt{2}\,x &= \sqrt{401} - 1 \\ \left( \sqrt{2}\,x \right)^2 &= \left( \sqrt{401} - 1 \right)^2 \\ 2x^2 &= 401 - 2\sqrt{401} + 1 \\ 2x^2 - 402 &= -2\sqrt{401} \\ x^2 - 201 &= -\sqrt{401} \\ x^2 &= 201 - \sqrt{401} \\ x&= \sqrt{201 - \sqrt{401}} \end{align*}$

So clearly $\displaystyle \begin{align*} \frac{\sqrt{401} - 1}{\sqrt{2}} = \sqrt{201 - \sqrt{401}} \end{align*}$ .

**Code2004** · January 20th 2013, 09:15 PM

Of course, excellent. Thanks!

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