Answer:
D
Step-by-step explanation:
We want to simplify the expression:
[tex]\displaystyle \frac{6i}{9+2i}[/tex]
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:
[tex]\displaystyle =\frac{6i}{9+2i}\left(\frac{9-2i}{9-2i}\right)[/tex]
Multiply:
[tex]\displaystyle = \frac{6i(9-2i)}{(9+2i)(9-2i)}[/tex]
Difference of two squares:
[tex]\displaystyle = \frac{6i(9-2i)}{(9)^2 -(2i)^2}[/tex]
Simplify:
[tex]\displaystyle = \frac{54i-12i^2}{(81)-(4i^2)} = \frac{54i-(-12)}{81-(-4)} = \frac{12+54i}{85}[/tex]
Hence, our answer is D.
