Question

${(i+\sqrt{3})}^{100}+{(i-\sqrt{3})}^{100}+2^{100}=$

• A. 1
• B. -1
• C. 0
• D. none of these

Answer 1

Answer: (C)

0

We have, $i+\sqrt{3}=\frac{-1+\sqrt{3}i}{2}.\frac{2}{i}=\frac{2\omega }{i}$

and $i-\sqrt{3}=\frac{-1-\sqrt{3}i}{2}.\frac{2}{i}=\frac{2{\omega }^2}{i}$

$\therefore {(i+\sqrt{3})}^{100}+{(i-\sqrt{3})}^{100}+2^{100}$

$={\left(\frac{2\omega }{i}\right)}^{100}+{\left(\frac{2{\omega }^2}{i}\right)}^{100}+2^{100}$

$=\frac{2^{100}}{i^{100}}({\omega }^{100}+{\omega }^{200})+2^{100}$

$=2^{100}(\omega +{\omega }^2)+2^{100}=-2^{100}+2^{100}=0$