Question

The real part of $(i - \sqrt{3} )^{13}$ is

• A. $2^{-10} $
• B. $2^{12} $
• C. $ 2^{-12} $
• D. $ - 2^{- 12} $
• E. $2^{10}$

Answer 1

Answer: (B)

$(i-\sqrt{3})^{13}$
$=2^{13} \times i^{13}\left[\frac{1+\sqrt{3} i}{2}\right]^{13}$
$=2^{13} i^{13}(-1)^{13}\left[\frac{-1-\sqrt{3} i}{2}\right]^{13}$
$=-2^{13} \cdot 1^{13} w^{13}=-2^{13} \cdot i \cdot\left[\frac{-1+\sqrt{3} i}{2}\right]$
$=-2^{13}[-i-\sqrt{3}]=-i 2^{13}+2^{13} \sqrt{3}$
Hence, real part is $2^{13} \sqrt{3}$.