Question

Simplify each of the following by rationalizing the denominator

A.$\frac{6-4\sqrt{2}}{6+4\sqrt{2}}$

B.$\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

C.$\frac{1}{3\sqrt{2}-2\sqrt{3}}$

D.$\frac{3\sqrt{5}-\sqrt{7}}{3\sqrt{3}+\sqrt{2}}$

Answer 1

A :

$=\frac{6-4\sqrt{4}}{6+4\sqrt{2}}\times \frac{6+4\sqrt{2}}{6+4\sqrt{2}}$

$=\frac{1}{17-12\sqrt{2}}$

B :

$=\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\times \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

$=\frac{2}{17+2\sqrt{35}}$

C :

$=\frac{1}{3\sqrt{2}-3\sqrt{3}}\times \frac{3\sqrt{2}+3\sqrt{3}}{3\sqrt{2}+3\sqrt{3}}$

$=\frac{3\sqrt{2}-2\sqrt{3}}{6}$

D :

$=\frac{3\sqrt{5}-7}{3\sqrt{3}+\sqrt{2}}\times \frac{3\sqrt{3}-\sqrt{2}}{3\sqrt{3}-\sqrt{2}}$

$=\frac{3\sqrt{15}-\sqrt{10}-2\sqrt{21}+\sqrt{14}}{29-6\sqrt{6}}$