This nasty little problem can be solved by Testing Cases, though it's useful first to try to understand what the math means. The question stem asks whether the sum of the cubes of two numbers is greater than the sum of the squares. It's pretty easy to think of yes examples:
if x = 2 and y = 3, then x^3 + y^3 > x^2 + y^2. What about no examples? Any negative numbers would yield a no answer, since the cube of a number would retain the negative sign but the square of a negative always turns positive. In addition, simple fractions (those between 0 and 1) get smaller when raised to a power, not larger.
(1) INSUFFICIENT. Keep these number properties in mind and test out the first statement: x + y > x^2 + y^2
Hmm. What next? Negatives will be invalid because statement (1) would be false: the squares will turn positive, which are larger by definition than negatives. Also note that since you've already found a no answer, the real question is whether you can ever find a yes answer.
It turns out you can, though the math is a bit messy. (One reason this question is harder than anything you'd see on the real test: you probably do need a calculator to reasonably complete it in 2 minutes.)
You need to identify a case in which squaring the two starting numbers decreases the sum but cubing them actually increases it. To get the first to occur, one of the numbers has to be a simple fraction (between 0 and 1). To get the second to occur, the second number needs to be larger enough than 1 to offset the decrease in the simple fraction but not so large that it makes x2 + y2 larger than x + y.
Try this:
For clarification: 1/4 + 36/25 = 169/100 or 1.69 and 1/8 + 216/125 = 1853/1000 or 1.853.
(2) INSUFFICIENT. Test some cases again.
The first case returned a yes. Can you find a way to get a no? The key difference is that the statement uses fourth powers while the question stem uses cubes. One key change, then, may be negative numbers. Add another row to your table and try another case:
(1) AND (2) INSUFFICIENT. You know the drill: test cases. To save yourself some effort, try things that you have already tried previously.
In the first case, above, try the pair that gave the "weird" result for statement (1). You've done most of the work already; all you have to do is check whether statement (2) is made true by these two numbers. It is, so now you have a yes answer to the question.
Now, can you get a no answer?
The fractional values used for statement (1) won't work for statement (2) (because taking those to the fourth power will make x^4 + y^4
smaller than x^2 + y^2). The positive and negative integers used for statement (2) won't work for statement (1). Try another weird pair like the 1/2 and 1.2 combo, but this time, move closer to 1 so that the simple fraction has more "weight" in the sum (that is, to give a better chance that the cubed sum winds up being less than the squared sum).
Even using both statements, there are yes and no answers to the question.
The correct answer is [spoiler](E)[/spoiler].
