In general, complex numbers are treated specially because they are the a squared number that is equal to a negative number. This isn't possible in traditional math because a positive times a positive and a negative times a negative both produce a positive.
This property is true.
[tex] \sqrt{x}^2 = x[/tex]
This property is also true.
[tex] \sqrt{x} * \sqrt{x} = \sqrt{x*x} [/tex]
We also know that [tex]x^2 = x * x[/tex]. The problem comes when you mix these two properties together. Lets solve each one practically and see what happens.
This is straight forward, plug and chug:
[tex]\sqrt{-6}^2 = -6[/tex]
This one takes some more work, but still comes out to a simple answer.
[tex]\sqrt{-6} * \sqrt{-6} = \sqrt{-6*-6}[/tex]
[tex]\sqrt{-6} * \sqrt{-6} = \sqrt{36}[/tex]
[tex]\sqrt{-6} * \sqrt{-6} = 6[/tex]
The problem is we have two different answers for the same definition. This contradiction is why complex number notation was created.
[tex]\sqrt{6}i[/tex] is how [tex]\sqrt{-6}[/tex] is written typically with the 6 part being the six from the radical and the i being [tex]\sqrt{-1}[/tex].
From this, we can multiply [tex]\sqrt{6}i[/tex] and [tex]\sqrt{6}i[/tex] to find the answer to your question.
[tex]\sqrt{6}i * \sqrt{6}i[/tex]
[tex]\sqrt{6}*\sqrt{6} * i^2[/tex]
[tex]6 * -1[/tex]
[tex]-6[/tex]
