Solving Fifth Roots of z = 1 + √2

[dogma]

Here's a silly roots question that has my congested mind temporarily stumped:

Let $$z = 1 + \sqrt{2}$$. Find the five distinct fifth roots of z.

Thanks in advance for helping me relieve the pressure.

 

[loandbehold]

Some of the roots are going to be complex, so the way I would tackle the problem is to rewrite your number in the form:

$$z=(1+\sqrt{2}){\rm e}^{2\pi ni}$$,

where n=0,1,2,... Then taking the fifth root gives:

$$z^{1/5}=(1+\sqrt{2})^{1/5} {\rm e}^{2\pi ni/5}$$,

which you can write in the form:

$$z^{1/5}=(1+\sqrt{2})^{1/5} \left \{ \cos \left ( \frac{2\pi n}{5} \right) +i \sin \left( \frac{2\pi n}{5} \right) \right \}$$.

Evaluating this for different n, should give 5 distinct roots.

 

[dogma]

thank you very much for the insight...I now proceed to kick myself for not seeing it on my own {sound of kicking}

Thanks!