Question

If $x+iy={\left(1-i\sqrt{3}\right)}^{100}$, then find $\left(x,y\right).$

• A. $\left(2^{99},2^{99}\sqrt{3}\right)$
• B. $\left(2^{99},-2^{99}\sqrt{3}\right)$
• C. $\left(-2^{99},2^{99}\sqrt{3}\right)$
• D. None of these

Answer 1

Answer: (C)

${\left(1-i\sqrt{3}\right)}^{100}=2^{100}{\left(-\frac{1}{2}+\frac{i\sqrt{2}}{2}\right)}^{100}$
$=2^{100}{\omega }^{100}$
$=2^{100}\omega$
$=2^{100}\left(-\frac{1}{2}+\frac{\sqrt{3}i}{2}\right)$
$=-2^{99}+2^{99}\sqrt{3}i$
Now, $x+iy={\left(1-i\sqrt{3}\right)}^{100}$
$=-2^{99}+2^{99}\sqrt{3}i$
$\Rightarrow x=-2^{99},y=2^{99}\sqrt{3}$
$\therefore \left(x,y\right)=\left(-2^{99},-2^{99}\sqrt{3}\right)$