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I'm having a bit of trouble with using the argument function with DeMoivres formula. I have the question:

z^{8}= 16

and am meant to find the solution using DeMoivre's formula (z^{n}=r^{n}(cosn(Θ) +isinn(Θ)) ). The problem is, I have no idea what an argument function is or how to find it. I've read around a bit and found thatroot(aand that^{2}+b^{2})= rtan^{-1}(b/a) =arg

but as you can see, I only have a (16) so that b=0, so is the argument then, tan^{-1}(0)? I have several problems and they all have a but no b. However, in the answer section, they all have arguments. What the heck am I doing wrong?

2. Relevant equations

I can do this one:

zas it has no argument (though I don't know why), and is just a matter of plugging in numbers.^{3}=-1

z^{3}=-1

z^{3}=1(cos(Θ)+isin(Θ)

z=_{3}√(1)(cos(Θ)+isin(Θ))^{1/3}

z=1(cos(2kπ)+isin(2kπ)^{1/3}

z=1(cos(2kπ/3)+isin(2kπ/3)

and then, since it has no argument, k can equal 0, +/-1, +/-2, +/-3 etc..., you plug in the k and solve.

3. The attempt at a solution

I haven't gotten very far with this one

z^{8}=16

z^{8}= 16^{8}(cos8(Θ)+isin8(Θ)) or if I do it the other way

z=_{8}√(16)(cos(2kπ)+isin(2kπ))^{1/8}

z=_{8}√(16)(cos(2kπ/8)+isin(2kπ/8))

and I can't insert anything for k, because I don't know the argument. I know I sound like a real novice at math (I am), but I hope someone can help me! Thank you so much in advance!

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# Homework Help: Using Argument and DeMoivre's theory

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