Euler's formulae

[Tommy_gun]

Could some please explain in detail (great detail), How to use eulers equation to solve, find all roots of, z^3=i
Thankyou

 

[pka]

Could some please explain in detail (great detail), How to use eulers equation to solve, find all roots of, z^3=i

First, express \(\displaystyle i=\exp\left(\frac{\pi}{2}\right)\) in polar form.

Second, raise that to the one-third power: \(\displaystyle \exp\left(\frac{\pi}{6}\right)\). That is one root.

Third, the roots are equally spaced on a circle. Thus we add \(\displaystyle \frac{2\pi}{3}\) to \(\displaystyle \frac{\pi}{6}\) twice to get the other two roots.

 

[Tommy_gun]

Ok how do I expand that to answer the question, z^5=3

 

[pka]

Ok how do I expand that to answer the question, z^5=3

\(\displaystyle 3=3\exp(0)\)
The fifth root of that is \(\displaystyle \sqrt[5]{3}\exp(0)\).
The angle dispersion is \(\displaystyle \frac{2\pi}{5}\) add that to 0 four more times.