Example. Find all values of $\sqrt[3]{-27 i}$.

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Solution.

1. The cube root of the complex number $z=-27 i$ has three different values, which are determined by the formula

$$
w=\sqrt[3]{-27 i}=\sqrt[3]{r} e^{i \frac{\varphi+2 \pi k}{3}}=\sqrt[3]{r}\left(\cos \frac{\varphi+2 \pi k}{3}+i \sin \frac{\varphi+2 \pi k}{3}\right)
$$

where $r=|-27 i|, \varphi=\arg (-27 i)$ and $k=0,1,2$.

2. We find the modulus and argument of the number $z=-27 i \quad(x=0, y=-27)$ :

$$
r=|-27 i|=27, \quad \varphi=\arg (-27 i)=-\pi / 2
$$

3. We find the values of the root $w_{0}, w_{1}, w_{2}$ using formula (2) for $k=0,1,2$ and represent them in algebraic form:

$$
\begin{aligned}
& w_{0}=3 e^{-\pi i / 6}=3\left(\cos \left(-\frac{\pi}{6}\right)+i \sin \left(-\frac{\pi}{6}\right)\right)=\frac{3 \sqrt{3}}{2}-\frac{3}{2} i \\
& w_{1}=3 e^{\pi i / 2}=3\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)=3 i \\
& w_{2}=3 e^{\frac{7 \pi i}{6}}=3\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right)=-\frac{3 \sqrt{3}}{2}-\frac{3}{2} i
\end{aligned}
$$

Answer. $\sqrt[3]{-27 i}$ has three values

$$
\frac{3 \sqrt{3}}{2}-\frac{3}{2} i, \quad 3 i, \quad-\frac{3 \sqrt{3}}{2}-\frac{3}{2} i
$$

Problem Conditions. Find all values of the root of the complex number.

1. $\sqrt[3]{1}$. 2. $\sqrt[3]{i}$. 3. $\sqrt[3]{-1} . \quad$ 4. $\sqrt[4]{-4} . \quad$ 5. $\sqrt{2+2 \sqrt{3}}$. 6. $\sqrt{-2+2 \sqrt{3}} i$. 7. $\sqrt{-2-2 \sqrt{3}} i$. 8. $\sqrt[6]{-27}$. 9. $\sqrt[6]{1}$. $10 . \sqrt[4]{-8-8 \sqrt{3} i}$.

Answers.

1. $1,(-1 \pm \sqrt{3} i) / 2 . \quad 2 .-i,( \pm \sqrt{3}+i) / 2 . \quad 3 .-1,(1 \pm \sqrt{3} i) / 2$. 4. $\pm(1 \pm i)$. 5. $\pm(\sqrt{3}+i)$. 6. $\pm(1+\sqrt{3} i)$. 7. $\pm(1-\sqrt{3} i)$. 8. $\pm \sqrt{3} i$, $\pm(3 \pm \sqrt{3} i) / 2$. $9 . \pm 1, \pm(1 \pm \sqrt{3} i) / 2$. $\quad 10 . \pm 1 \pm \sqrt{3} i$.

## 1.2. Curves in the Complex Plane

Problem Statement. Determine the type of curve defined by the equation $z(t)=x(t)+i y(t), t \in(-\infty, \infty)$.

Plan of Solution.

1. Form the parametric equations of the curve

$$
\left\{\begin{array}{l}
x=x(t) \\
y=y(t), \quad t \in(-\infty, \infty)
\end{array}\right.
$$

where $x(t)=\operatorname{Re} z(t)$ and $y(t)=\operatorname{Im} z(t)$.

2. By eliminating the parameter $t$ from the parametric equations, we obtain the equation of the curve in the form

$$
F(x, y)=0
$$

3. Using the canonical forms of equations of curves in the plane, determine the type of the desired curve.
4. Find the ranges of $x(t)$ and $y(t)$ and determine which part of the curve is defined by the original equation.

Remark. The point with coordinates $x(t), y(t)$ may traverse the curve multiple times.