How to Calculate the Complex Roots of x^5 = 10?

[mardybum9182]

Homework Statement

Given x^5=10, find the five roots and plot them on an Argand diagram.

Relevant Equations

Newton's method.

Mentor note: Member reminded that some effort must be shown.
Real root is 1.858.
Just don't know which method to use to find the 4 complex roots

 

[anuttarasammyak]

$$x^5=10$$
$$(\frac{x}{\sqrt[5]{10}})^5=z^5=1$$

So get solutions of $z^5=1$ and multiply them with $\sqrt[5]{10}$.

 

[FactChecker]

Newton's method will only find the real solutions. With complex numbers, you should consider the properties of complex numbers to get the other, complex solutions. Even for the real solution, they probably don't want you to use Newton's method, but rather just leave that solution as $10^{1/5}$.
For the complex solutions, do you know about the representation of a complex number in its polar form ($re^{i\theta}$)? If so, consider 10 in its polar form: $10e^{i0} = 10e^{i2\pi}= 10e^{i4\pi}= 10e^{i6\pi}= 10e^{i8\pi}= 10e^{i10\pi}$. Now take the fifth root of the individual factors, 10 and $e^{i2n\pi}$

 

[Vanadium 50]

Just don't know which method to use to find the 4 complex roots

Do you know what an Argand diagram is?