Question

Simplify. 6i/9+2i A. 81-18i/85 B. 2i/3 C. 6i/13 D. 12+54i/85

Answer 1

Answer:

D

Step-by-step explanation:

We want to simplify the expression:

$\displaystyle \frac{6i}{9+2i}$

To do so, we can remove the imaginary unit in the denominator by multiply it by the conjugate.

The conjugate of a + bi is a - bi.

Hence, we will multiply the fraction by 9 - 2i:

$\displaystyle =\frac{6i}{9+2i}\left(\frac{9-2i}{9-2i}\right)$

Multiply:

$\displaystyle = \frac{6i(9-2i)}{(9+2i)(9-2i)}$

Difference of two squares:

$\displaystyle = \frac{6i(9-2i)}{(9)^2 -(2i)^2}$

Simplify:

$\displaystyle = \frac{54i-12i^2}{(81)-(4i^2)} = \frac{54i-(-12)}{81-(-4)} = \frac{12+54i}{85}$

Hence, our answer is D.