To get the 4th roots, let's convert first the complex number to its polar form. Here are the steps:
1. Find the absolute value of r.
[tex]r=\sqrt{a^2+b^2}^[/tex]In the given complex number, a = 2√3 while b = -2.
[tex]\begin{gathered} r=\sqrt{(2\sqrt{3})^2+(-2)^2} \\ r=\sqrt{12+4} \\ r=\sqrt{16} \\ r=4 \end{gathered}[/tex]2. Solve for the measure of the angle theta.
Since a > 0, we can use the tangent function to solve for the angle (in radian).
[tex]\begin{gathered} \theta=tan^{-1}\frac{b}{a} \\ \theta=tan^{-1}\frac{-2}{2\sqrt{3}} \\ \theta=-\frac{\pi}{6} \end{gathered}[/tex]Now that we have r = 4 and θ = -π/6, let's now determine the roots using the formula below:
[tex]\alpha=\sqrt[4]{r}(cos(\frac{\theta}{4})+isin(\frac{\theta}{4}))[/tex]Plugin the value of r and θ in the formula above.
[tex]\begin{gathered} \alpha=\sqrt[4]{4}(cos(\frac{-\frac{\pi}{6}}{4})+isin(\frac{-\frac{\pi}{6}}{4})) \\ \alpha=\sqrt[4]{4}(cos(-\frac{\pi}{24})+isin(-\frac{\pi}{24}) \\ \alpha=\sqrt[4]{4}cis(-\frac{\pi}{24}) \end{gathered}[/tex]To determine the next root, simply add π/2 to the angle.
[tex]\begin{gathered} \alpha=\sqrt[4]{4}cis(-\frac{\pi}{24}+\frac{\pi}{2}) \\ \alpha=\sqrt[4]{4}cis(\frac{11\pi}{24}) \end{gathered}[/tex]Add another π/2.
[tex]\begin{gathered} \alpha=\sqrt[4]{4}cis(\frac{11\pi}{24}+\frac{\pi}{2}) \\ \alpha=\sqrt[4]{4}cis(\frac{23\pi}{24}) \end{gathered}[/tex]Last, add another π/2.
[tex]\begin{gathered} \alpha=\sqrt[4]{4}cis(\frac{23\pi}{24}+\frac{\pi}{2}) \\ \alpha=\sqrt[4]{4}cis(\frac{35\pi}{24}) \end{gathered}[/tex]The roots in order of increasing angle measure are:
[tex]\begin{gathered} 1:\sqrt[4]{4}cis(-\frac{\pi}{24}) \\ 2:\sqrt[4]{4}cis(\frac{11\pi}{24}) \\ 3:\sqrt[4]{4}cis(\frac{23\pi}{24}) \\ 4:\sqrt[4]{4}cis(\frac{35\pi}{24}) \end{gathered}[/tex]
