Question

Prove that ${\left(1+i\right)}^4{\left(1+\frac{1}{i}\right)}^4=16$

Answer 1

$LHS$

${(1+i)}^4{(1+\frac{1}{i})}^4$

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

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

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

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

$=2i\times 2i\times \frac{2}{i}\times \frac{2}{i}$

$=16$

$RHS$

Hence, proved.