Question

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

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is C $0$

${\left(i+\sqrt{3}\right)}^{100}+{\left(1-\sqrt{3}\right)}^{100}$

${\left({\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^2\right)}^{100}+{\left(\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)\times 2\right)}^{100}$

$2^{100}\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)+2^{100}{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^{100}$

$2^{100}{\left(\frac{\cos \pi }{6}+\frac{i\sin \pi }{6}\right)}^{100}+\cos \left(\frac{-\pi }{6}\right)+i\sin {\left(\frac{-\pi }{6}\right)}^{100}+1$

$\left(e^{\frac{i50\pi }{3}}+e^{-i{50}^{\frac{\pi }{3}}}+1\right)$

$\left(2\cos \left(\frac{50\pi }{3}\right)+1\right)=2^{100}\left(2\cos \left(\frac{2\pi }{2}\right)+1\right)$

$\text{=}{\text{2}}^{100}\left(0\right)$