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

$(a)\;2i\qquad(b)\;i\qquad(c)\;0\qquad(d)\;-i$

Answer : $\;i$
Explanation :
$i^{-39} = i^{-4 \times 9-3}$
$= (i^{4})^{-9} \;. i^{-3}$
$= (1)^{-9} \; . i^{-3} \qquad [i^{4} =1]$
$= \large\frac{1}{i^{3}} = \large\frac{1}{-i} \qquad [i^{3} =-i]$
$= -\large\frac{1}{i} \times \large\frac{i}{i}$
$= \large\frac{-i}{i^2}$
$= \large\frac{-i}{-1} \qquad [i^{2} =-1]$
$= i$