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irina [24]
3 years ago
15

Solve using demoivres theorem

Mathematics
1 answer:
pashok25 [27]3 years ago
3 0
It's difficult to make out, but I think the task is to expand

(\sqrt3-i)^4

Write the number in polar form first:

\sqrt3-i=2e^{-i\pi/6}

By DeMoivre's theorem, you have

(\sqrt3-i)^4=\left(2e^{-i\pi/6}\right)^4=2^4e^{-i4\pi/6}=16e^{-i2\pi/3}

and converting back to Cartesian form, this number is equivalent to

16\left(\cos\dfrac{2\pi}3-i\sin\dfrac{2\pi}3\right)=16\left(-\dfrac12+-\dfrac{\sqrt3}2\right)=-8(1+i\sqrt3)
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see the attached figure to better understand the problem

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