Question

Express the mentioned below complex number in the standard form $a+ib$
(iii) $\frac{1}{(2+i{)}^2}$

Answer 1

$\because \frac{1}{(2+i{)}^2}$
$=\frac{1}{2^2+i^2+2\times 2\times i}$
$=\frac{1}{4+(-1)+4i}$
$=\frac{1}{3+4i}$
$=\frac{1}{3+4i}\times \frac{3-4i}{3-4i}$
$=\frac{3-4i}{3^2-(4i{)}^2}$
$=\frac{3-4i}{25}$
$=\frac{3}{25}-\frac{4}{25}i$
$\therefore \frac{1}{(2+i{)}^2}=\frac{3}{25}-\frac{4}{25}i$