# Express the given complex number in the form a + ib: i–39

Question:

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

Solution:

$i^{-39}=i^{-4 \times 9-3}=\left(i^{4}\right)^{-9} \cdot i^{-3}$

$=(1)^{-9} \cdot i^{-3} \quad\left[i^{4}=1\right]$

$=\frac{1}{i^{3}}=\frac{1}{-i} \quad\left[i^{3}=-i\right]$

$=\frac{-1}{i} \times \frac{i}{j}$

$=\frac{-i}{i^{2}}=\frac{-i}{-1}=i \quad\left[i^{2}=-1\right]$