help with simplification (1 Viewer)

atifaj · Feb 1, 2020

Simplify

fan96 · Feb 1, 2020

Let
$x = \sqrt{a+\sqrt b} - \sqrt{a-\sqrt b}$ .

Now,
$\begin{aligned} x^2 &= (a + \sqrt b) - 2\sqrt{ (a + \sqrt b) (a - \sqrt b)} + (a - \sqrt b) \\ &= 2(a - \sqrt{a^2-b}).\end{aligned}$

Clearly $x \ge 0$ , so

$x = \sqrt{2(a - \sqrt{a^2-b})}.$

Set $a = 11, \, b = 2$ and you're done.

atifaj · Feb 1, 2020

fan96 said:
Let
$x = \sqrt{a+\sqrt b} - \sqrt{a-\sqrt b}$ .

Now,
$\begin{aligned} x^2 &= (a + \sqrt b) - 2\sqrt{ (a + \sqrt b) (a - \sqrt b)} + (a - \sqrt b) \\ &= 2(a - \sqrt{a^2-b}).\end{aligned}$

Clearly $x > 0$ , so

$x = \sqrt{2(a - \sqrt{a^2-b})}.$

Set $a = 11, \, b = 2$ and you're done.

Thanks a lot

help with simplification (1 Viewer)

atifaj

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fan96

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atifaj

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