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.
![\sqrt{x}^2 = x](https://tex.z-dn.net/?f=%20%20%5Csqrt%7Bx%7D%5E2%20%3D%20x)
This property is also true.
![\sqrt{x} * \sqrt{x} = \sqrt{x*x}](https://tex.z-dn.net/?f=%20%5Csqrt%7Bx%7D%20%2A%20%20%5Csqrt%7Bx%7D%20%20%3D%20%20%5Csqrt%7Bx%2Ax%7D%20)
We also know that
![x^2 = x * x](https://tex.z-dn.net/?f=x%5E2%20%3D%20x%20%2A%20x)
. 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:
![\sqrt{-6}^2 = -6](https://tex.z-dn.net/?f=%5Csqrt%7B-6%7D%5E2%20%3D%20-6)
This one takes some more work, but still comes out to a simple answer.
![\sqrt{-6} * \sqrt{-6} = \sqrt{-6*-6}](https://tex.z-dn.net/?f=%5Csqrt%7B-6%7D%20%2A%20%20%5Csqrt%7B-6%7D%20%20%3D%20%20%5Csqrt%7B-6%2A-6%7D)
![\sqrt{-6} * \sqrt{-6} = \sqrt{36}](https://tex.z-dn.net/?f=%5Csqrt%7B-6%7D%20%2A%20%20%5Csqrt%7B-6%7D%20%20%3D%20%20%5Csqrt%7B36%7D)
![\sqrt{-6} * \sqrt{-6} = 6](https://tex.z-dn.net/?f=%5Csqrt%7B-6%7D%20%2A%20%20%5Csqrt%7B-6%7D%20%20%3D%206)
The problem is we have two different answers for the same definition. This contradiction is why complex number notation was created.
![\sqrt{6}i](https://tex.z-dn.net/?f=%5Csqrt%7B6%7Di)
is how
![\sqrt{-6}](https://tex.z-dn.net/?f=%5Csqrt%7B-6%7D)
is written typically with the 6 part being the six from the radical and the i being
![\sqrt{-1}](https://tex.z-dn.net/?f=%5Csqrt%7B-1%7D)
.
From this, we can multiply
![\sqrt{6}i](https://tex.z-dn.net/?f=%5Csqrt%7B6%7Di)
and
![\sqrt{6}i](https://tex.z-dn.net/?f=%5Csqrt%7B6%7Di)
to find the answer to your question.
![\sqrt{6}i * \sqrt{6}i](https://tex.z-dn.net/?f=%5Csqrt%7B6%7Di%20%2A%20%5Csqrt%7B6%7Di)
![\sqrt{6}*\sqrt{6} * i^2](https://tex.z-dn.net/?f=%5Csqrt%7B6%7D%2A%5Csqrt%7B6%7D%20%2A%20i%5E2)