Question

Express the given complex number in the form $a+ib:$$i^{-39}$

Answer 1

$i^{-39}=i^{4\times (-9)-3}={\left(i^4\right)}^{-9}.i^{-3}$
$={\left(1\right)}^{-9}.i^{-3},[\because i^4=1]$
$=\frac{1}{i^3}=\frac{1}{-i},[\because i^3=-i]$
$=\frac{-1}{i}\times \frac{i}{i}$
$=\frac{-i}{i^2}=\frac{-i}{-1}=i,[\because i^2=-1]$
$=a+ib$, where $a=0,b=1$