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

$=\;\left[\left(-1\right)^9\;+\frac1{i^{24}.i}\right]^3\;=\;\left[-1\;+\frac1{\left(i^2\right)^{12}.i}\right]^3$
$=\;\left[-1\;+\frac1{\left(-1\right)^{12}.i}\right]^3\;=\;\left[-1\;+\frac1{1\times i}\right]^3$
$=\left[-1+\frac1i\right]^3\;=\;\left[-1-i\right]^3$
$=-\left(1\;+i\right)^3\;=\;-\left[1+i^3+3\times1\times i(1+i)\right]$
$=-\left[1-i+3i(1+i)\right]\;=\;-\left[1-i+3i+3i^2\right]$
$=-\left[1-i+3i-3\right]\;=\;-\left[-2+2i\right]$
$=2\;-\;2i$

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