Ian Stewart wrote:

Mo2men wrote: Can I square both sides in question stem as follows?

No, you cannot do that - in general, if you have an inequality, if one side of the inequality might be negative, it is not correct to square both sides. For example, if you start with this inequality, which is clearly true:

-3 < 2

then if you square both sides, you'll get something that is false (9 is not less than 4). So squaring both sides of an inequality is mathematically illegal (except when you know both sides are positive). Here, the right side of the inequality is certainly positive, but the left side, |a| - |b|, might be negative. So squaring is not correct.

If you know how to interpret absolute values as distances (I don't have time to explain that method now, but I've posted about it several times in the past on this forum, so you might find an explanation with a search), you can answer this question fairly quickly. The answer to the question is usually "yes", and is only "no" when b is in between a and zero. In that case, so if either a < b < 0 or 0 < b < a, the left and right sides of the inequality in the question are equal. Since Statement 2 tells us b is not in between a and zero, it is sufficient. Statement 1 is a bit problematic (the expression b^a can be undefined if a is a non-integer) but it only tells us b is negative, and tells us nothing important about where a is, so Statement 1 is not sufficient.