Question

Simplify by rationalizing the denominator.
$\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

• A. $\frac{114-40\sqrt{6}}{30}$
• B. $\frac{104-41\sqrt{6}}{30}$
• C. $\frac{114-41\sqrt{6}}{3}$
• D. $\frac{114-41\sqrt{6}}{30}$

Answer 1

The correct option is D $\frac{114-41\sqrt{6}}{30}$

$\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$
The conjugate of $\sqrt{48}+\sqrt{18}$ is $\sqrt{48}-\sqrt{18}$

$=\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{48}+\sqrt{18}}\times$$\frac{\sqrt{48}-\sqrt{18}}{\sqrt{48}-\sqrt{18}}$

$=\frac{7\sqrt{48\times 3}-7\sqrt{3\times 18}-5\sqrt{48\times 2}+5\sqrt{18\times 2}}{48-18}$

$=\frac{7\times 12-7\times 3\sqrt{6}-5\times 4\sqrt{6}+5\times 6}{30}$

$=\frac{114-41\sqrt{6}}{30}$