Rationalize-the-denominator-of-2-1-2-4-1-3-

Question Number 59626 by naka3546 last updated on 12/May/19

R a t i o n a l i z e t h e d e n o m i n a t o r o f

\frac{2}{1 - \sqrt{2 + \sqrt[3]{4}}}

Answered by MJS last updated on 12/May/19

first use this formula

\frac{1}{a - \sqrt{b}} = \frac{a + \sqrt{b}}{(a - \sqrt{b}) (a + \sqrt{b})} = \frac{a + \sqrt{b}}{a^{2} - b}

then use this formula

\frac{1}{a + \sqrt[3]{b}} = \frac{(a + \sqrt[3]{b} (- \frac{1}{2} - \frac{\sqrt{3}}{2} i)) (a + \sqrt[3]{b} (- \frac{1}{2} + \frac{\sqrt{3}}{2} i))}{(a + \sqrt[3]{b}) (a + \sqrt[3]{b} (- \frac{1}{2} - \frac{\sqrt{3}}{2} i)) (a + \sqrt[3]{b} (- \frac{1}{2} + \frac{\sqrt{3}}{2} i))} =

= \frac{a^{2} - a \sqrt[3]{b} + \sqrt[3]{b^{2}}}{a^{3} + b}

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