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If $x+i y=(1-i \sqrt{3})^{100}$, then find $(x, y)$.
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1395 Upvotes
Verified Answer
The correct answer is:
$\left(-2^{99}, 2^{99} \sqrt{3}\right)$
$\quad \therefore \mathrm{ROR}^{-1}=\{(3,3),(3,5),(5,3),(5,5)\}$ $(1-\mathrm{i} \sqrt{3})^{100}=2^{100}\left(-\frac{1}{2}+\frac{\mathrm{i} \sqrt{3}}{2}\right)^{100}$ $=2^{100} \omega^{100}=2^{100} \omega$ $=2^{100}\left(-\frac{1}{2}+\frac{\sqrt{3} \mathrm{i}}{2}\right)=-2^{99}+2^{99} \sqrt{3} \mathrm{i}$
$$
\begin{array}{l}
\text { Now, } x+i y=(1-i \sqrt{3})^{100} \\
=-2^{99}+2^{99} \sqrt{3} i \\
\Rightarrow x=-2^{99}, y=2^{99} \sqrt{3} \\
\therefore(x, y)=\left(-2^{99},-2^{99} \sqrt{3}\right)
\end{array}
$$
$$
\begin{array}{l}
\text { Now, } x+i y=(1-i \sqrt{3})^{100} \\
=-2^{99}+2^{99} \sqrt{3} i \\
\Rightarrow x=-2^{99}, y=2^{99} \sqrt{3} \\
\therefore(x, y)=\left(-2^{99},-2^{99} \sqrt{3}\right)
\end{array}
$$
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