How do you simplify #(2i)^(1/2)#?
1 Answer
There are two answers:
and
Explanation:
Consider
Then
Therefore, equating real and imaginary parts separately for left and right sides of this equation, we get a system of two equations with two unknowns
or, simplifying,
From the first equation we conclude that either

If
#x=y# , from the second equation follows that
#x^2=1# and either#x=1# or#x=1#
So, we have two solutions:
#sqrt(2i) = 1+i# or#sqrt(2i) = 1i#
Check:
#(1+i)^2 = 1+2i+i^2 = 1+2i1 = 2i# (GOOD)
#(1i)^2 = (1)^2+2i+(i)^2 = 1+2i1 = 2i# (GOOD) 
If
#x=y# , from the second equation follows that
#x^2=1# , which has no solutions among real numbers.