Question

Evaluate ${\left(\frac{1+i}{\sqrt{2}}\right)}^{8n}-{\left(\frac{1-i}{\sqrt{2}}\right)}^{8n}=$

• A. $0$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $0$

${\left(\frac{1+i}{\sqrt{2}}\right)}^{8n}-{\left(\frac{1-i}{\sqrt{2}}\right)}^{8n}$

$=\frac{\left\{\right(1+i{)}^2{\}}^{4n}}{(\sqrt{2}{)}^{8n}}-\frac{\left\{\right(1-i{)}^2{\}}^{4n}}{(\sqrt{2}{)}^{8n}}$

$=\frac{\{2i{\}}^{4n}}{2^{4n}}-\frac{\{-2i{\}}^{4n}}{2^{4n}}$ [Since $(1+i{)}^2=2i$ and $(1-i{)}^2=-2i$ and $i^2=-1$]

$=i^{4n}-(-i{)}^{4n}$

$=1-1=0$. [ Since $i^4=1$]