If ab >0, is (ab)^2 < (ab)^(1/2) ?

edit: the OP fixed a typo in the original post above, so this solution is to the question I have quoted above (with the typo in it) and not to the now-edited question at the top of the page (which I solve in a separate post below)

First, if we replace "ab" with "x", the question is asking:

Is x^2 <√x ?

and x^2 < √x is only true if 0 < x < 1, so the question is asking "Is ab < 1?"

Statement 1 tells us:

4/a > 7ab

Before determining if this is sufficient, notice that we know ab > 0. So the right side of this inequality is positive, which means 4/a is greater than some positive number, and therefore a itself is positive. So Statement 1 guarantees that a is positive. But that's the only information you get from Statement 2. So you don't learn anything new by combining the two statements that you don't know from Statement 1 on its own, and the OA provided cannot be correct -- the answer can only be A or E, since Statement 2 alone is clearly not sufficient.

Using Statement 1 alone (which we can rewrite, since a is positive, as 4 > 7a^2 b or as 4/7 > a^2 b if we want to), it's easy to see that ab < 1 is possible, using the values a = 0.1 and b = 1, say. But it's also easy enough to find values where ab > 1. We can take, say, a = 0.1 and b = 11.

So we can't say if ab < 1 is true, and the answer is E unless there's a typo in the question.