Elimination of radials - Confused¿?

Take some numbers to understand this.
The equality sign means that whatever is on the left hand side, it is equal to whatever there is on the right hand side. So if the LHS is 2, RHS is also 2

2 = 2
Now, you can square both sides and the equality will still hold 2^2 = 2^2
You can raise both sides to any power, the equality will hold.

You can also multiply the same number on both the sides, the inequality will still hold 2*3 = 2*3

You can choose to do whichever operation suits your purpose. Since you want to get rid of the root sign, you would want to square both sides.

Also, notice that here √(3b-8)^2 = √(12-b) * √(3b-8),
√(3b-8) = √(12-b) (even though they don't look same but they are since it is given to us) so √(3b-8)^2 = √(12-b) * √(3b-8) is the same as √(3b-8)^2 = √(12-b)*√(12-b)