Question

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

Answer 1

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

=$[i^{4×4+2}+(\frac{1}{i^{4×4+1}})]^3$

=$[i^{(4)^4.i^2}+\frac{1}{(i^{4})^6.i}]^3$

$=[i^2+\frac{1}{i}]^3$ $[i^{-4}=1]$

$=[-1+\frac{1}{i}×\frac{1}{i}]^3$ $[i^4=-1]$

$=[-1+\frac{i}{i^2}]^3$

$=[-1-i]^3$

=$(-1)^3[1+i]^3$

$=[1^3+i^3+3.1.i(1+i)]$

$=-[1+i^3+3i+3i^2]$

$=-[1+i^3+3i+3i^2]$

$=-[1-i+3i-3]$

$=-[-2+2i]$

$=2-2i$