if i=√-1,then √i=?

Question

Choose correct answer given below - (A). -1, (B) ±1-i√2,(c) ±1+i/2,(D) 1±i

Answer 1

I think some of the answer options are incorrect.

Let a+ib=√i, then a²+2aib-b²=i, where a and b are both real.

So, equating real and imaginary parts, 2ab=1 and a²-b²=0, b=±a.

±2a²=1. Therefore, since a is real, a=1/√2 which is the same as √2/2.

If a=1/√2, then b=1/√2 because 2ab>0 (product of a and b must be positive).

Hence √i (implying the positive square root) is (1+i)/√2.

However, if the square root is taken to be positive or negative then:

Square root of i=±(1+i)/√2, or ±(1+i)√2/2, which is not covered by any of the answer options.

CHECK FOR VALIDITY

[±(1+i)/√2]²=(1+2i-1)/2=i✔️