You are correct to say there is two solutions here
Your book has given 'a' root, not all.
Question. Find a square root of the complex number .
Answer.
Let . By inspection (of i), the modulus of w is 1; we need to find .
where n=2 and k is 0 and 1 (from 0 to n-1).
Therefore there are two numbers w that, when squared, give :
and
The textbook answer, however, only gives one value of w ( ). Am I wrong in thinking that there are two values of w? Or, the question has only asked for one of them ('a square root'), rather than 'find all square roots'?
PS I understand that there are always two numbers (one positive and one negative) that, when squared, give the same number. I only wanted to check whether the same principle applies to the questions about complex numbers.