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. $2-2i$

Answer 1

Answer: (E)

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