Apparent contradiction during complex analysis

Help me understand if I have made a mistake somewhere in the following calculations:

-(2i/k) * sqrt[ω(-ω')] = -(2i/k) * sqrt[ω(-1)ω'] = -(2i/k) * sqrt(-1) * sqrt[ωω'] = -(2i/k) * i * sqrt[ωω'] = + (2/k)sqrt(ωω')

but also:

-(2i/k) * sqrt[ω(-ω')] = -(2/k) * sqrt[(i^2)ω(-ω')] = -(2/k) * sqrt[(-1)ω(-ω')] = -(2/k)sqrt(ωω')

How can both equalities hold at the same time, even though the signs are opposite?