Question

Express the following in $a+ib$ form
$\frac{4i^8-3i^9+3}{3i^{11}-4i^{10}-2}$

Answer 1

$\frac{4i^8-3i^9+3}{3i^{11}-4i^{10}-2}$

$=\frac{4-3i+3}{-3i+4-2}$ [Since $i^4=1\Rightarrow i^8=1$ and $i^9=i$ and $i^{10}=-1$]

$=\frac{7-3i}{2-3i}$

$=\frac{(7-3i)(2+3i)}{2^2-(3i{)}^2}$

$=\frac{(7-3i)(2+3i)}{13}$ [Since $i^2=-1$]

$=\frac{(23+15i)}{13}$

$=\frac{23}{13}+i\frac{15}{23}$

This is the required $a+ib$ from.