95/VB02. problem / Math - Solved Math Problems

MR-95 / 230b. problem

Simplify the following expression:

$\sqrt{6 - 4 \sqrt{2}}$

$\sqrt{6 - 4 \sqrt{2}}$

= \sqrt{6 - \sqrt{4^{2} 2}}

$= \sqrt{6 - \sqrt{32}}$

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

$a = 6; b = 32$

$\sqrt{6 - \sqrt{32}}$

= \sqrt{\frac{6 + \sqrt{6^{2} - 32}}{2}} - \sqrt{\frac{6 - \sqrt{6^{2} - 32}}{2}}

= \sqrt{\frac{6 + \sqrt{36 - 32}}{2}} - \sqrt{\frac{6 - \sqrt{36 - 32}}{2}}

= \sqrt{\frac{6 + \sqrt{4}}{2}} - \sqrt{\frac{6 - \sqrt{4}}{2}}

= \sqrt{\frac{6 + 2}{2}} - \sqrt{\frac{6 - 2}{2}}

$= \sqrt{\frac{8}{2}} - \sqrt{\frac{4}{2}}$

= \sqrt{4} - \sqrt{2}

$= 2 - \sqrt{2}$

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