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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}

 Feb 18, 2024
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avatar+1622 
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\(\frac{1}{\sqrt{36}} - \sqrt{27} - \frac{1}{\sqrt{27}} - \sqrt{18} + \frac{1}{\sqrt{18}} - \sqrt{9}\)

Begin by simplifying the radicals: \({1\over6}-3\sqrt{3}-{1\over{3\sqrt{3}}}-3\sqrt{2}+{1\over{3\sqrt{2}}}-3\)

Then, rationalize the denominators: for example, \({1\over\sqrt{x}}={1\over\sqrt{x}}*{\sqrt{x}\over\sqrt{x}}={\sqrt{x}\over{x}}\):

\({1\over{6}}-3\sqrt{3}-{3\sqrt{3}\over27}-3\sqrt{2}+{3\sqrt{2}\over18}-3\)

Simplify the denominators, then combine like terms: \({1\over{6}}-3\sqrt{3}-{\sqrt{3}\over9}-3\sqrt{2}+{\sqrt{2}\over6}-3\)

Put everyone under the same common denominator, 18: \({3\over{18}}-{54\sqrt{3}\over18}-{2\sqrt{3}\over18}-{54\sqrt{2}\over18}+{3\sqrt{2}\over18}-{54\over18}\)

Combine like terms: \({-51-56\sqrt{3}-51\sqrt{2}\over18}\), I trust that you can split up the fraction if the answer specifies so.

 Feb 19, 2024

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