Perform the following operation and write the result in standard form.
$$\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }$$
A) $\frac { 9 } { 40 } + \frac { 83 } { 80 } i$
B) $\frac { 27 } { 10 } + \frac { 83 } { 10 } i$
C) $\frac { 1 } { 41 } + \frac { 9 } { 82 } i$
D) $\frac { 27 } { 82 } + \frac { 83 } { 82 } i$
E) $\frac { 27 } { 8 } + \frac { 83 } { 8 } i$

Question

Quizplus · Accepted Answer

To simplify the given expression, we need to rationalize the denominators of both fractions. For the first fraction, multiply the numerator and denominator by the conjugate of the denominator, $9 - i$, and for the second fraction, multiply by the conjugate of its denominator, $9 + i$. First fraction:$$\frac{9i}{9+i} \cdot \frac{9-i}{9-i} = \frac{9i(9-i)}{9^2 - i^2} = \frac{81i - 9i^2}{81 + 1} = \frac{81i + 9}{82}$$Second fraction:$$\frac{2}{9-i} \cdot \frac{9+i}{9+i} = \frac{2(9+i)}{9^2 + i^2} = \frac{18 + 2i}{81 + 1} = \frac{18 + 2i}{82}$$Adding the two results:$$\frac{81i + 9}{82} + \frac{18 + 2i}{82} = \frac{27 + 83i}{82}$$Thus, the result in standard form is $\frac{27}{82} + \frac{83}{82}i$, which matches choice D.

Perform the Following Operation and Write the Result in Standard 9i9+i+29−i\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }9+i9i​+9−i2​

Perform the Following Operation and Write the Result in Standard $\frac { 9 i } { 9 + i } + \frac { 2 } { 9 - i }$