psm12se, could you please publish the source of your question? This question is not in a GMAT format, although it tests principles that you'll see on real GMAT problems. A real GMAT problem, though, would always say "which of the following MUST be true?"
Hemant, you are correct in deducing that X must be greater than 0, because it is larger than something squared. Beyond that, though, we don't know much of anything.
When we're looking at inequalities with exponents, we have to consider several things: positives/negatives, and integers/fractions. We aren't given any contraints for X, Y, and Z, so we don't know what kind of numbers they are.
We know that even exponents hide the sign of their base - the base could be positive or negative, but the square will always be positive. So let's say that X=10, Y^2=9, and Z^4=1. This satisfies X > Y^2 > Z^4. But Y could be 3 or -3, and Z could be 1 or -1. There is no way to tell whether Y is greater than Z, so we can't claim that any of the statements must be true. We can disprove all of them.
Even if we had been given a constraint that X, Y, and Z were positive, we still wouldn't be able to answer the question. Let's say that X=(1/3), Y^2=(1/4), and Z^4=(1/16). This satisfies X > Y^2 > Z^4. But here, X=(1/3), but Y and Z are both equal to (1/2). We can't tell whether X is greater than Y or Z.
As Hemant said, we can think of situations that can be true for each statement, but none of the statements must be true.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
