Question

Evaluate ${[i^{18}+{\left(\frac{1}{i}\right)}^{25}]}^3$

Answer 1

Given data
(let) $z={[i^{18}+{\left(\frac{1}{i}\right)}^{25}]}^3$
Using exponent property
$={[i^{4\times 4+2}+\frac{1}{i^{4\times 6+1}}]}^3$
$={\left[\right(i{)}^2+\frac{1}{i}]}^3$ $\{\therefore i^4=1\}$
Arrange in numerator form
$={\left[\right(i{)}^2+\frac{1}{i}\times \frac{i}{i}]}^3$
$={\left[\right(i{)}^2+\frac{i}{i^2}]}^3$ $\because \{i^2=-1\}$
$={[-1-i]}^3$
$z=[i^{18}+(\frac{1}{i}{)}^{25}{]}^3=[-1-i{]}^3$
$=-(1+i{)}^3$
Apply $(A+B{)}^3$ formula
$=-\left[\right(1{)}^3+(i{)}^3+3(1{)}^2\left(i\right)+3\left(1\right)\left(i{)}^2\right]$
$=-[1+i^3+3i+3i^2]$
$=-[1-i+3i-3]$
$=-[-2+2i]$
$=2-2i$