I encountered this problem in my exercise set, and was stumped.
Solve (z + 1)^4 = 1 - i
My approach to the problem was to recognize that (1 - i) could be rewritten as [ (1 - i)^4 ] / 8.
Which then gives..
(z + 1)^4 = (1 - i)^4 / 8
(z + 1) = +/- (1 - i) / 4throot(8)
But I'm not sure how to proceed from here, since it would appear that in solving each of the resulting two equations, I would only obtain two roots, rather than the expected four.
Edit: Sorry, but LaTeX kept giving me an error when I tried using it.
Hmm...I like this solution! Thank you!
Our text stipulates, however, that we should follow the "outline" used in one of the examples; i.e. with some clever algebraic manipulation put the complex polynomial equation in a solvable Cartesian form (i.e. do not use the polar form of the complex numbers). When I attempted it this way, I got stuck as I mentioned.
In any case, I think it's pretty dumb when in a few steps you can see (much more clearly at that) what the roots are.
I do have one question, though...is it acceptable to leave the solution in a non-standard form? i.e. the polar form with a "-1" tacked at the end? Or should you expand the polar form into its trigonometric representation and go through the grueling task of algebraically rewriting the expression in a standard form? If it is acceptable...then is it possible in this non-standard form to plot the roots (by inspection) on the complex plane, or is this pretty much an "analytic only" exact answer?