This is a simple proof that each real number undergoing a normal operation is equal to a real number.

[tex]\sqrt{-1} = \sqrt{-(1)} = \sqrt{-(\sqrt{1})^{2}}[/tex]

Since,

[tex]\sqrt{x^{y}} = \sqrt{x}^{y}[/tex]

Then,

[tex]\sqrt{-(\sqrt{1})^{2}} = \sqrt{-(\sqrt{1})}^{2}[/tex]

Thus,

[tex]\sqrt{-1} = -\sqrt{1} = -1[/tex]

Or generally,

[tex]\sqrt{-n} = -\sqrt{n}[/tex]