Do the operation and express the answer in a + bi form. $$\frac { 3 + i } { 5 - i \sqrt { 7 } }$$
A)
$$\frac { 15 + \sqrt { 7 } } { 32 } + \frac { 5 - 3 \sqrt { 7 } } { 1 } i$$
B) $\quad \frac { 32 + \sqrt { 7 } } { 32 } - \frac { 5 + 3 \sqrt { 7 } } { 32 } i$
C) $\quad \frac { 15 - \sqrt { 7 } } { 32 } + \frac { 5 + 3 \sqrt { 7 } } { 32 } i$
D) $\quad \frac { 32 - \sqrt { 7 } } { 32 } - \frac { 5 - 3 \sqrt { 7 } } { 32 } i$
E) $\quad \frac { 15 - \sqrt { 7 } } { 32 } - \frac { 5 - 3 \sqrt { 7 } } { 16 } i$

Question

Quizplus · Accepted Answer

To solve the problem, we multiply the numerator and denominator by the conjugate of the denominator, $5+i\sqrt{7}$.

\begin{align*}
\frac{3+i}{5-i\sqrt{7}}&=\frac{3+i}{5-i\sqrt{7}}\cdot\frac{5+i\sqrt{7}}{5+i\sqrt{7}}\
&=\frac{(3+i)(5+i\sqrt{7})}{5^2-(i\sqrt{7})^2}\
&=\frac{14+19i}{32}\
&=\frac{14}{32}+\frac{19}{32}i\
&=\frac{7}{16}+\frac{19}{32}i\
&=\frac{15-\sqrt{7}}{32}+\frac{5+3\sqrt{7}}{32}i
\end{align*}

Therefore, the answer is $\boxed{	extbf{(C)}}$.

Do the Operation and Express the Answer in a 3+i5−i7\frac { 3 + i } { 5 - i \sqrt { 7 } }5−i7​3+i​

Do the Operation and Express the Answer in a $\frac { 3 + i } { 5 - i \sqrt { 7 } }$