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Simplify the expression \[\frac{6}{\sqrt{36} + \sqrt{27}} + \frac{6}{\sqrt{27} + \sqrt{18}} + \frac{6}{\sqrt{18} + \sqrt{9}}.\]

 Jun 15, 2019
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\(\ \phantom{=\quad}\frac{6}{\sqrt{36} + \sqrt{27}} + \frac{6}{\sqrt{27} + \sqrt{18}} + \frac{6}{\sqrt{18} + \sqrt{9}}\\~\\~\\ =\quad\Big(\frac{6}{\sqrt{36} + \sqrt{27}}\cdot\frac{\sqrt{36}-\sqrt{27}}{\sqrt{36}-\sqrt{27}}\Big) + \Big(\frac{6}{\sqrt{27} + \sqrt{18}}\cdot\frac{\sqrt{27}-\sqrt{18}}{\sqrt{27}-\sqrt{18}}\Big) + \Big(\frac{6}{\sqrt{18} + \sqrt{9}}\cdot\frac{\sqrt{18}-\sqrt9}{\sqrt{18}-\sqrt9}\Big)\\~\\~\\ =\quad\frac{6\sqrt{36}-6\sqrt{27}}{36-27}+ \frac{6\sqrt{27}-6\sqrt{18}}{27-18} + \frac{6\sqrt{18}-6\sqrt9}{18-9}\\~\\~\\ =\quad\frac{6\sqrt{36}-6\sqrt{27}}{9}+ \frac{6\sqrt{27}-6\sqrt{18}}{9} + \frac{6\sqrt{18}-6\sqrt9}{9}\\~\\~\\ =\quad\frac{6\sqrt{36}-6\sqrt9}{9}\\~\\~\\ =\quad\frac{36-18}{9}\\~\\~\\ =\quad2 \)_

 Jun 15, 2019

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