Question

I need help with this practiceThe subject is complex numbers and vectors*It asks to fill in the four boxes, *enter the roots in order of increasing angle measure

Answer 1

ANSWER:

$\begin{gathered} \:\sqrt[4]{4}cis\left(-\frac{\pi }{24}\right)\: \\ \\ \:\sqrt[4]{4}cis\left(\frac{11\pi}{24}\right)\: \\ \\ \:\sqrt[4]{4}cis\left(\frac{23\pi}{24}\right)\: \\ \\ \:\sqrt[4]{4}cis\left(\frac{35\pi}{24}\right)\: \end{gathered}$

STEP-BY-STEP EXPLANATION:

We have the following expression:

$2\sqrt{3}-2i$

To calculate the 4 roots we must match the equation with x raised to 4, just like this:

$x^4=2\sqrt{3}-2i$

For this case the roots are given as follows:

$\begin{gathered} \:z_k=\sqrt[n]{\left|a\right|}\left(\cos\left(\frac{\arctan\left(\alpha\right)+2k\pi}{n}\right)+i\sin\left(\frac{\arctan\left(\alpha\right)+2k\pi}{n}\right)\right) \\ \\ \text{ In this case:} \\ \\ n=4 \\ \\ |a|=\sqrt{\left(2\sqrt{3}\right)^2+\left(-2\right)^2}=\sqrt{12+4}=\sqrt{16} \\ \\ a=4 \\ \\ \alpha=\left(\frac{-2}{2\sqrt{3}}\right) \\ \\ \text{ Therefore:} \\ \\ \arctan\left(\frac{-2}{2\sqrt{3}}\right)=-\frac{\pi}{6} \end{gathered}$

Taking into account the above, we calculate for each x,

when k = 0,1, 2 3, just like this:

$\begin{gathered} x_1=\sqrt[4]{4}\left(\cos\left(\frac{-\frac{\pi}{6}+2\cdot\:0\pi}{4}\right)+i\sin\left(\frac{-\frac{\pi}{6}+2\cdot\:0\pi}{4}\right)\right)=\sqrt[4]{4}\left(\cos\left(-\frac{\pi}{24}\frac{}{}\right)+i\sin\left(-\frac{\pi}{24}\right)\right)=\sqrt[4]{4}cis\left(-\frac{\pi}{24}\right) \\ \\ x_2=\sqrt[4]{4}\left(\cos\left(\frac{-\frac{\pi}{6}+2\cdot\:1\pi}{4}\right)+i\sin\left(\frac{-\frac{\pi}{6}+2\cdot\:1\pi}{4}\right)\right)=\sqrt[4]{4}\left(\cos\left(\frac{11\pi\:}{24}\right)+i\sin\left(\frac{11\pi\:}{24}\right)\right)=\:\sqrt[4]{4}cis\left(\frac{11\pi\:}{24}\right)\: \\ \\ x_3=\sqrt[4]{4}\left(\cos\left(\frac{-\frac{\pi}{6}+2\cdot\:2\pi}{4}\right)+i\sin\left(\frac{-\frac{\pi}{6}+2\cdot\:2\pi}{4}\right)\right)=\sqrt[4]{4}\left(\cos\left(\frac{23\pi\:}{24}\right)+i\sin\left(\frac{23\pi\:}{24}\right)\right)=\sqrt[4]{4}cis\left(\frac{23\pi\:}{24}\right)\: \\ \\ x_4=\sqrt[4]{4}\left(\cos\left(\frac{-\frac{\pi}{6}+2\cdot\:3\pi}{4}\right)+i\sin\left(\frac{-\frac{\pi}{6}+2\cdot\:3\pi}{4}\right)\right)=\sqrt[4]{4}\left(\cos\left(\frac{35\pi\:}{24}\right)+i\sin\left(\frac{35\pi\:}{24}\right)\right)=\:\sqrt[4]{4}cis\left(\frac{35\pi\:}{24}\right)\: \end{gathered}$