Question

The value of $\left( i^{18} + \left( \frac{1}{i} \right)^{25} \right)^3$ is equal to

• A. $\frac{1 + i}{2}$
• B. $ 2 + 2i $
• C. $\frac{1 - i }{2} $
• D. $\sqrt{2} - \sqrt{2} i $
• E. $ 2 - 2 i$

Answer 1

Answer: (E)

$\left\{i^{18}+\left(\frac{1}{i}\right)^{25}\right.\}^{3}=\left\{\left(i^{4}\right)^{4} \cdot i^{2}+\left(\frac{1}{i^{4}}\right)^{6} \cdot \frac{1}{i}\right\}^{3}$
$=\left[1 \cdot(-1)+1 \cdot \frac{1}{i}\right]^{3}$
$=\left[\frac{1}{i}-1\right]^{3}$
$=\frac{1}{i^{3}}-1+\frac{3}{i}\left(1-\frac{1}{i}\right)$
$=i-1-3 i+3$
$=2-2 i$