Question

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

Answer 1

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

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

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

$={(i^2+\frac{1}{i})}^3$ ($\because i^4=1$)

$={(-1+\frac{1}{i}\times \frac{i}{i})}^3$ ($\because i^2=-1$)

$={(-1+\frac{i}{i^2})}^3$

$={(-1-i)}^3$

$={(-1)}^3{(1+i)}^3$

$=-[1^3+i^3+\mathrm{3.1.}i(1+i)]$

$=-(1+i^3+3i+3i^2)$

$=-(1-i+3i-3)$

$=-(-2+2i)$

$=2-2i$