For squares we have following properties.
1. If the number is less than 0, the square is always greater than the number
2. If the number is between 0 and 1, the square is always smaller than the number
3. If the number is greater than 1, the square is always greater than the number.

In case of inequalities,

If the numbers lie between 0 & 1, then we have to reverse the direction of inequality.
i.e. if a and b both are between 0 & 1, and a^2 > b^2 then a <b> b^2 then a <b> b^2 then a > b

simplyjat wrote: If the numbers lie between 0 & 1, then we have to reverse the direction of inequality.

I think you must have meant something else because this isn't true.

Ex: .5 > .25 and (.5)^2=.25 > (.25)^2=.0625

The rule for squaring inequalities is this:

If a > b and |a| > |b|, then a^2 > b^2.
If a > b and |a| < |b|, then a^2 < b^2.

My advice is to never take the square root of inequalities.

If you have x^2 > 9 for example, taking the square root would give you x > 3. But that is only part of the solution, since there is also x < -3.

Instead of taking the square root, do the following:
1. Pretend that the inequality is an equals sign, so x^2 = 9.
2. Find the solutions.
3. Think about the behavior of the inequality around those points. In this case, with -3 and 3, you should realize that with numbers above 3 or below -3, the square is greater than 9. With numbers in between, the result is less than 9.

Squaring is a bit different, but you should still be careful.
If root(x) < 4, we can't just square and say that x < 16.
This is because we also have the limitation that we can only take the square root of a positive number or zero. So the solution would be 0 <= x < 16.

In either case, I would advise you to try to reason through the question rather than taking the square or square root. If you have any specific examples in mind, please share them and I'd be happy to make more specific suggestions.

(1) Suppose a > b > 0

Multiply (1) by a to get: a^2 > ab
Multiply (1) by b to get: ab > b^2
Put the above together to get: a^2 > ab > b^2

This is because we also have the limitation that we can only take the square root of a positive number or zero.

This is not true. The square root of -1 is equal to i (some notations use j). Complex numbers....

bobybobbobbobybob wrote:

This is because we also have the limitation that we can only take the square root of a positive number or zero.

This is not true. The square root of -1 is equal to i (some notations use j). Complex numbers....

All numbers on the GMAT are real.
Thus, on the GMAT, it is not possible to take an even root of a negative value.

bobybobbobbobybob wrote:

This is because we also have the limitation that we can only take the square root of a positive number or zero.

This is not true. The square root of -1 is equal to i (some notations use j). Complex numbers....

This is not relevant to the GMAT, which doesn't recognize the existence of imaginary and complex numbers. (We're pre-Renaissance here.)