Tell me how this happens?

It's the 2nd to the last step (meaning from the end result of 1) involving the square.

That creates an ambiguity that can go either way and the result of which is unreliable. So it is chosen to select the incorrect outcome emphasizing the contradictory result.

But

(-1)^6 = 1

Therefore, -1 IS a sixth root of 1.

Here's another example using same logic

a = -2

a^2 = 4

√(a^2) = √4

a = 2

You can not multiply negative numbers under an even-rooted radical !

example:

√(-1) * √(-4) = i * 2i = 2i^2 = - 2

√[(- 1)*(- 4)] = √[(1)(4)(-1)(-1)] = 2i^2 = - 2

NOT WHAT YOU DID with (- 1)^2 under a sixth rooted radical.

QED

The error is in the 5th term where you create (-1^2)^1/6

It's a fallacy. You have to simplify the exponent before proceeding with applying it to the base.

This is pretty cool but there is small problem with it. As soon as you square any side of an equation you have to square both sides.