Simplify the expression \[\frac{6}{\sqrt{36} + \sqrt{27}} + \frac{6}{\sqrt{27} + \sqrt{18}} + \frac{6}{\sqrt{18} + \sqrt{9}}.\]
simplify | 6/(sqrt(36) + sqrt(27)) + 6/(sqrt(27) + sqrt(18)) + 6/(sqrt(18) + sqrt(9))
6/[6 + 3sqrt(3)] + 6/[3sqrt(3) + 3sqrt(2)] + 6/[3sqrt(2) + 3]
2/[sqrt(3) + 2] + 2/[sqrt(3) + sqrt(2)] + 2/[sqrt(2) + 1]
4 - 2 sqrt(3) - 2 sqrt(2) + 2 sqrt(3) + 2 sqrt(2) - 2
= 2
+26367 Simplify the expression \[\frac{6}{\sqrt{36} + \sqrt{27}} + \frac{6}{\sqrt{27} + \sqrt{18}} + \frac{6}{\sqrt{18} + \sqrt{9}}.\]
\(\begin{array}{|rcll|} \hline && \frac{6}{\sqrt{36} + \sqrt{27}} + \frac{6}{\sqrt{27} + \sqrt{18}} + \frac{6}{\sqrt{18} + \sqrt{9}} \\\\ &=& 6\cdot \left( \frac{1}{\sqrt{36} + \sqrt{27}} + \frac{1}{\sqrt{27} + \sqrt{18}} + \frac{1}{\sqrt{18} + \sqrt{9}} \right)\\\\ &=& 6\cdot \left( \frac{\sqrt{36}-\sqrt{27}}{(\sqrt{36}+\sqrt{27})(\sqrt{36}-\sqrt{27})} + \frac{\sqrt{27}-\sqrt{18}}{(\sqrt{27}+\sqrt{18})(\sqrt{27}-\sqrt{18})} + \frac{\sqrt{18}-\sqrt{9}}{(\sqrt{18}+\sqrt{9})(\sqrt{18}-\sqrt{9})} \right)\\\\ &=& 6\cdot \left( \frac{\sqrt{36}-\sqrt{27}}{36-27} + \frac{\sqrt{27}-\sqrt{18}}{27-18} + \frac{\sqrt{18}-\sqrt{9}}{18-9} \right)\\\\ &=& 6\cdot \left( \frac{\sqrt{36}-\sqrt{27}}{9} + \frac{\sqrt{27}-\sqrt{18}}{9} + \frac{\sqrt{18}-\sqrt{9}}{9} \right)\\\\ &=& 6\cdot \left( \frac{\sqrt{36}}{9} -\underbrace{\frac{\sqrt{27}}{9} + \frac{\sqrt{27}}{9} }_{=0} -\underbrace{\frac{\sqrt{18}}{9} + \frac{\sqrt{18}}{9} }_{=0} -\frac{\sqrt{9}}{9} \right)\\\\ &=& 6\cdot \left( \frac{\sqrt{36}}{9} -\frac{\sqrt{9}}{9} \right)\\\\ &=& \frac{6}{9}\cdot \left( \sqrt{36} - \sqrt{9} \right)\\\\ &=& \frac{6}{9}\cdot \left( 6-3 \right)\\\\ &=& \frac{6}{9}\cdot 3 \\\\ &=& \frac{6}{3} \\\\ &\mathbf{=}& \mathbf{2} \\ \hline \end{array}\)
