Question

Simplify the following:

${\displaystyle {\text{i}}^{59}+\frac{1}{{\text{i}}^{59}}}$

Answer 1

${\displaystyle {\text{i}}^{59}+\frac{1}{{\text{i}}^{59}}={\text{i}}^{4\left(14\right)+3}+\frac{\text{i}}{{\text{i}}^{4\left(14\right)+3}}}$

= ${\displaystyle {\text{i}}^3+\frac{1}{{\text{i}}^3}}$

= ${\displaystyle -\text{i}+\frac{1\times \text{i}}{-\text{i}\times \text{i}}}$

= – i + i

= 0