Simplify by rationalizing the denominator. Write your answer in a +bi form.
-$\frac { 7 + 4 i } { 7 + 5 i }$
A) $\frac { 29 } { 24 } + \frac { 7 } { 24 } \mathrm { i }$
B) $\frac { 29 } { 74 } - \frac { 63 } { 74 } \mathrm { i }$
C) $\frac { 23 } { 8 } + \frac { 7 } { 24 } \mathrm { i }$
D) $\frac { 69 } { 74 } - \frac { 7 } { 74 } \mathrm { i }$

Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $7+5i$ is $7-5i$. Thus, we have:

$$\frac{7+4i}{7+5i} \cdot \frac{7-5i}{7-5i} = \frac{(7)(7) + (7)(-5i) + (4i)(7) + (4i)(-5i)}{49+25} = \frac{69-7i}{74}$$

Therefore, the answer is $\frac{69}{74} - \frac{7}{74}i$, which is choice D.

Simplify by Rationalizing the Denominator 7+4i7+5i\frac { 7 + 4 i } { 7 + 5 i }7+5i7+4i​

Simplify by Rationalizing the Denominator $\frac { 7 + 4 i } { 7 + 5 i }$