is non-negative, so we can just move some stuff around:
Say for example we considered something like . It's possible for to be a negative number, but if it was negative, the value on the left-hand side would have been negative, and a negative number can't be greater than 1000. So we just need to make a side note that we are only considering values of such that . From then we can continue the problem.
As for your second question: Any time you take an even-powered root of a value, you must take its absolute value. In other words:
Notice that had we not done so, we would not have found the solution , but clearly .
You do not need to do this when taking odd-powered roots.