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Simplify the expression
\frac{1}{\sqrt{36}} - \sqrt{27} - \frac{1}{\sqrt{27}} - \sqrt{18} + \frac{1}{\sqrt{18}} - \sqrt{9}

 Nov 11, 2023
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We simplify the expression by simplifying each radical term separately. [ \begin{aligned} \frac{1}{\sqrt{36}} - \sqrt{27} - \frac{1}{\sqrt{27}} - \sqrt{18} + \frac{1}{\sqrt{18}} - \sqrt{9} =& \frac{1}{6} - 3\sqrt{3} - \frac{1}{3\sqrt{3}} - 3\sqrt{2} + \frac{1}{3\sqrt{2}} - 3\ =& \frac{1}{6} - 3(\sqrt{3} + \sqrt{2}) + \frac{1}{3}(\frac{1}{\sqrt{3}} + \frac{1}{\sqrt{2}}) - 3\ =& \frac{1}{6} - 3(\sqrt{3} + \sqrt{2}) + \frac{1}{3}(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2}\sqrt{3}}) - 3\ =& \frac{1}{6} - 3(\sqrt{3} + \sqrt{2}) + \frac{\sqrt{2} + \sqrt{3}}{3} - 3\ =& \frac{1}{6} + \frac{\sqrt{2} + \sqrt{3}}{3} - 3 - 3(\sqrt{3} + \sqrt{2})\ =& \boxed{\frac{1}{6} + \frac{\sqrt{2} + \sqrt{3}}{3} - 6(\sqrt{3} + \sqrt{2})} \end{aligned} ]

 Nov 11, 2023

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