Chapter 9.3, Problem 64AYU

Use the result of Problem 65 to draw the conclusion that each complex $n$th root lies on a circle with center at the origin. What is the radius of this circle?

To determine

To show: Each complex nth root lies on a circle with center at the origin. What is the radius of this circle?

Answer to Problem 64AYU

Solution:

For a positive integer the complex number has exactly $n$ distinct $n$th roots given by $z_k=\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+ i\sin \frac{\theta +2\pi k}{n}\right)$, where $k=0,1,2\dots$ Here magnitude is $\sqrt[n]{r}$ and angle is $\frac{\theta +2\pi k}{n}$.

$\left|z_k\right|=\sqrt[n]{r}$, which is a circle with center $\left(0, 0\right)$ and radius is $\sqrt[n]{r}$.

Explanation of Solution

For a positive integer the complex number has exactly $n$ distinct $n$th roots given by $z_k=\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+ i\sin \frac{\theta +2\pi k}{n}\right)$, where $k=0,1,2\dots$ Here magnitude is $\sqrt[n]{r}$ and angle is $\frac{\theta +2\pi k}{n}$.

$\left|z_k\right|=\sqrt[n]{r}$, which is a circle with center $\left(0, 0\right)$ and radius is $\sqrt[n]{r}$.