Simplify by rationalizing the denominator.
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Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator which is $\sqrt{c^2 + d^2}$. 
So, 
$\frac{a+bi}{c-di} \cdot \frac{c+di}{c+di} = \frac{(a+bi)(c+di)}{c^2 + d^2}$ 
$= \frac{ac + adi + bci + bdi^2}{c^2 + d^2}$ 
$= \frac{(ac - bd) + (ad + bc)i}{c^2 + d^2}$ 
Therefore, the answer is $\frac{(11ebb63d_1bff_a116_b851_8960d484b136)_{16}}{(c^2+d^2)_{16}}$ which is option C. The given expression has a square root in the denominator, so we need to rationalize it. To do so, we multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate is $\sqrt{851} - 32$. 
\begin{align*}
\frac{1}{\sqrt{851} + 32} &= \frac{1}{\sqrt{851} + 32} \cdot \frac{\sqrt{851} - 32}{\sqrt{851} - 32}\
&= \frac{\sqrt{851} - 32}{851 - 32^2} \
&= \frac{\sqrt{851} - 32}{c828}
\end{align*}
Therefore, the answer is (C).

Simplify by Rationalizing the Denominator