Rationalize the denominator in the following expression. (Assume all expressions appearing under a square root symbol represent nonnegative numbers.)

Question

Quizplus · Accepted Answer

The denominator has the form $\sqrt{a+b\sqrt{c}}$, which can be rationalized by multiplying the expression by its conjugate, $\sqrt{a-b\sqrt{c}}$, in the numerator and denominator. Doing so gives $\frac{\sqrt{3}+\sqrt{2}\sqrt{5+2\sqrt{3}}}{\sqrt{3}-\sqrt{2}\sqrt{5+2\sqrt{3}}}$. To simplify this further, we multiply the numerator and denominator by the conjugate of the denominator, $\sqrt{3}+\sqrt{2}\sqrt{5+2\sqrt{3}}$, which gives $\frac{(\sqrt{3}+\sqrt{2}\sqrt{5+2\sqrt{3}})^2}{3-(2\sqrt{5+2\sqrt{3}})}$. Expanding the numerator gives $3+2\sqrt{3}+10+4\sqrt{3}=13+6\sqrt{3}$. Plugging this into the simplified expression gives $\frac{13+6\sqrt{3}}{1}=13+6\sqrt{3}$. Therefore, the answer is choice D.

Rationalize the Denominator in the Following Expression