Evaluate [ i^ 18 + ( 1 i ) ^ 25 ]^3 - Vidyadip

Question

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

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Answer

$=\left[(-1)^9+\frac{1}{i^{24} \cdot i}\right]^3=\left[-1+\frac{1}{\left(i^2\right)^{12} \cdot i}\right]^3$
$=\left[-1+\frac{1}{(-1)^{12} \cdot i}\right]^3=\left[-1+\frac{1}{1 \times i}\right]^3$
$=\left[-1+\frac{1}{i}\right]^3=[-1-i]^3$
$=-(1+i)^3=-\left[1+i^3+3 \times 1 \times i(1+i)\right]$
$=-[1-i+3 i(1+i)]=-\left[1-i+3 i+3 i^2\right]$
$=-[1-i+3 i-3]=-[-2+2 i]$
$=2-2 i$

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