Question

rationalize the denominator of 5 root 3 minus 3 root 2 divided by root 108 + root 18​

Answer 1

SOLUTION

TO DETERMINE

Rationalize the denominator

$\displaystyle \sf{ \frac{5 \sqrt{3} - 3 \sqrt{2} }{ \sqrt{108} + \sqrt{18} } }$

EVALUATION

$\displaystyle \sf{ \frac{5 \sqrt{3} - 3 \sqrt{2} }{ \sqrt{108} + \sqrt{18} } }$

$\displaystyle \sf{ = \frac{5 \sqrt{3} - 3 \sqrt{2} }{6 \sqrt{3} + 3 \sqrt{2} } }$

$\displaystyle \sf{ = \frac{(5 \sqrt{3} - 3 \sqrt{2})(6 \sqrt{3} - 3 \sqrt{2}) }{(6 \sqrt{3} + 3 \sqrt{2} )(6 \sqrt{3} - 3 \sqrt{2}) } }$

$\displaystyle \sf{ = \frac{90 - 15 \sqrt{6} - 18 \sqrt{6} + 18 }{{(6 \sqrt{3} )}^{2} - {(3 \sqrt{2})}^{2} } }$

$\displaystyle \sf{ = \frac{108 - 33 \sqrt{6} }{ 108 - 18}}$

$\displaystyle \sf{ = \frac{3(36 - 11 \sqrt{6} \: ) }{ 90}}$

$\displaystyle \sf{ = \frac{(36 - 11 \sqrt{6} \: ) }{30}}$

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Answer 2

$\rm{\displaystyle \frac{5\sqrt{3}-3\sqrt{2}}{\sqrt{108}+\sqrt{18}}}$

$\rm{\displaystyle =\frac{5\sqrt{3}-3\sqrt{2}}{6\sqrt{3}+3\sqrt{2}}}$

• Multiply by the Conjugate $\rm{\dfrac{6\sqrt{3}+3\sqrt{2}}{6\sqrt{3}+3\sqrt{2}}}$

$\rm{\displaystyle =\frac{\left(5\sqrt{3}-3\sqrt{2}\right)\left(6\sqrt{3}-3\sqrt{2}\right)}{\left(6\sqrt{3}+3\sqrt{2}\right)\left(6\sqrt{3}-3\sqrt{2}\right)}}$

$\rm{\displaystyle =\frac{108-33\sqrt{6}}{90}}$

• Factor $\rm{108-33\sqrt{6}}$

$\rm{\displaystyle =\frac{3\left(36-11\sqrt{6}\right)}{90}}$

• Cancel the Common Factor 3

$\rm{\displaystyle =\frac{36-11\sqrt{6}}{30}}$