# roots

• Sep 17th 2012, 09:34 AM
franios
roots
Compute all the 4th roots of a=1+√3 & describe where they are located in the complex plane.
• Sep 17th 2012, 09:49 AM
emakarov
Re: roots
According to the formula from another problem, [r , θ]ⁿ = [rⁿ, nθ]. If [r⁴, 4θ] = [1+√3, 0], then $r=\sqrt[4]{1+\sqrt{3}}$. You need to find all 0 ≤ θ < 2π such that 4θ is an integer multiple of 2π.
• Sep 17th 2012, 09:59 AM
Plato
Re: roots
Quote:

Originally Posted by franios
Compute all the 4th roots of a=1+√3 & describe where they are located in the complex plane.

Do you mean $a=1+\sqrt3~i~?$
• Sep 17th 2012, 02:12 PM
franios
Re: roots
Yes
• Sep 17th 2012, 02:18 PM
emakarov
Re: roots
Well, the idea is the same, only now $r=\sqrt[4]{|a|}$ and you need to find all 0 ≤ θ < 2π such that 4θ equals arg(a) up to an integer multiple of 2π
• Sep 17th 2012, 02:26 PM
Plato
Re: roots
Quote:

Originally Posted by franios
Yes

Then $a = 2\exp \left( {\frac{{i\pi }}{6}} \right)$.

The four fourth roots are ${\alpha _k} = \sqrt[4]{2}\exp \left( {\frac{{i\pi }}{{24}} + \frac{{i\pi k}}{2}} \right),~k=0,1,2,3$