Complex solutions to $a = (z+b)^n$

Question

I have tried the whole afternoon trying to figure out how to approach an equation of the form $a = (z+b)^n$, more specifically the equation: $1 = (z+1)^4$. Is there a general approach to equations of this form?

Math level: 1st year at university.

Answer 1

Generally: $$a=(z+b)^n\Rightarrow \left (\dfrac{z+b}{a^{1/n}}\right )^n=1\Rightarrow w^n=1$$ So $$w=e^{\dfrac{2k\pi i}{n}},k=0,1,2\dots n-1$$ In your case: $$(z+1)^4=1\Rightarrow z+1=e^\dfrac{2k\pi i}{4}, k=0,1,2,3$$