When you see the √ symbol, it always specifies the principal (non-negative) square root. In this case, in order to show that the intended result is negative, you have to specify that you want the *negative* square root by adding a minus sign in front.
√4 = 2
If you wanted the negative square root, you'd write:
-√4 = -2
Or if you wanted both, you'd write:
±√4 = -2 or 2
As noted in the problem, if x is negative, then x^3 is negative. But x^6 will be positive and so will the principal square root. To make it negative as it should be, they noted that x^3 = -√(x^6), when x is a negative value.