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 $2^{13}\sqrt{3}$.