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The modulus and amplitude of $(1+i \sqrt{3})^{8}$ are respectively
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Verified Answer
The correct answer is:
256 and $\frac{2 \pi}{3}$
Let $z=(1+i \sqrt{3})^{8}$
$=\left(2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)\right)^{8}$
$=\left[2\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\right]^{8}=\left(2^{8} e^{i \pi / 3}\right)^{8}$
$=2^{8} \cdot e^{\frac{8 \pi}{3} i}=2^{8} \cdot e^{\left(2 \pi+\frac{2 \pi}{3}\right) i}=2^{8} \cdot e^{\frac{2 \pi}{3} i}$
So, modulus $=2^{8}=256$ and amplitude $=\frac{2 \pi}{3}$
$=\left(2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)\right)^{8}$
$=\left[2\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\right]^{8}=\left(2^{8} e^{i \pi / 3}\right)^{8}$
$=2^{8} \cdot e^{\frac{8 \pi}{3} i}=2^{8} \cdot e^{\left(2 \pi+\frac{2 \pi}{3}\right) i}=2^{8} \cdot e^{\frac{2 \pi}{3} i}$
So, modulus $=2^{8}=256$ and amplitude $=\frac{2 \pi}{3}$
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