Mathematics: Explain your answer and i ll rate/mark the brainliest! which expression is a fourth root of -1 + i√3? answer choices below:

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Explain your answer and i ll rate/mark the brainliest!

marelinatalia2000

11.06.2020

$\sqrt[4]{2}(\cos 30\degree +i\sin 30\degree)$

Step-by-step explanation:

Let

$z=-1+i\sqrt{3}$

Then;

$r=|z|=\sqrt{(-1)^2+(\sqrt{3})^2 }=2$

Then

$Arg(z)=\tan^{-1}(\frac{\sqrt{3} }{-1})=120\degree$

Hence the given complex number polar form is

$z=2(\cos 120\degree +i\sin 120\degree)$

$(-1+i\sqrt{3})^{\frac{1}{4} }=(2(\cos 120\degree +i\sin 120\degree)) ^{\frac{1}{4} }$

We now use the DeMoivres Formula;

$((r(\cos \theta +i\sin \theta))^{\frac{p}{q}}$

$=r^{\frac{p}{q}}( \cos \frac{p}{q} \theta +i\sin \frac{p}{q} \theta)$

$((2(\cos 120\degree +i\sin 120\degree))^{\frac{1}{4}}$

$=2^{\frac{1}{4}}( (\cos \frac{1}{4}(120\degree) +i\sin \frac{1}{4}(120\degree))$

$=\sqrt[4]{2}( (\cos 30\degree) +i\sin 30\degree)$

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