Question

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

(Given: n is a natural number)

• A. $0$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

Answer: (C)

$2$

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

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

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

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

$=1+1$

$=2$