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complex numbers

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fredrick:
if z=1+i is a solution of of equation of the form z^4=a where a is a complex number. find all the other solutions??

Mao:
so, let

we can also express in polar form, but since is only one (of the four) solutions, we will stick with pronumerals until we arrive at the general solution



from De Moivre's theorem





we dont need to worry about , as we can work that out from



from , we can easily obtain that , assuming this is the first root where





substituting this, also taking into account that we have four roots here:









Converting back:



ps as you can see here, there's a very simple pattern, the simplistic approach (that i might be bothered to write)


a simplistic approach:

-> there are four roots.

therefore each root are away from each other.

the given is in the first quadrant, rewritten as

therefore, adding/subtracting successive gives:

Ahmad:


or

Let

dcc:








Using de'moivres:



The 4 solutions to this will all have a modulus of

Let m =

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