Question

Express the mentioned below complex number in the standard form $a+ib$
(v) $\frac{(2+i{)}^3}{2+3i}$

Answer 1

$\because \frac{(2+i{)}^3}{2+3i}$

$=\frac{2^3+i^3+3\times 2^2\times i+3\times 2\times i^2}{2+3i}$

$=\frac{8-i+12i+6i^2}{2+3i}$

$=\frac{8+11i+6\times (-1)}{2+3i}$

$=\frac{2+11i}{2+3i}\times \frac{2-3i}{2-3i}$

$=\frac{4-6i+22i-33i^2}{2^2-(3i{)}^2}$

$=\frac{37+16i}{13}$

$=\frac{37}{13}+\frac{16}{13}i$

$\therefore \frac{(2+i{)}^3}{2+3i}=\frac{37}{13}+\frac{16}{13}i$