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