complex plane

[lukelowe]

I can't figure out how to do these problems. For instance, find the principle fifth root of 2-5i. Where do I even start?

 

[tkhunny]

De Moivre?

 

[lukelowe]

Yes, but how do use the theory.

 

[Deleted member 4993]

I can't figure out how to do these problems. For instance, find the principle fifth root of 2-5i. Where do I even start?

\(\displaystyle a \ + \ ib \ = \ r\cdot e^{i(\theta + 2n\pi)}\)

\(\displaystyle [a \ + \ ib]^m \ = \ [r\cdot e^{i(\theta + 2n\pi)}]^m \ = \ r^n \cdot e^{i\cdot (m\theta + 2mn\pi)}\)

so a start would be to find r and ? for your problem

 

[lukelowe]

The only thing I can figure is r= radical 29 after that I have no clue. this crappy book I have doesn't do a very good job of explaining and my teacher rushed through it to get every thing done for the quarter.

 

[pka]

I can't figure out how to do these problems.
For instance, find the principle fifth root of 2-5i. Where do I even start?

Let \(\displaystyle \phi = \arctan \left( {\frac{{ - 5}}{2}} \right)\).
The principal fifth root of 2-5i is \(\displaystyle \sqrt[{10}]{{29}}\exp \left( {\frac{{\phi i}}{5}} \right)\).