This was the question:

Which is greater, x²*y² or x²+y²?

Professor Homunculus' answer:

I am thinking about an example from a GRE (graduate record exam) book that was shown to me.
I think it was "Which is greater, x²+y² or (x+y)²?

Here is the poop on how to think about examples like that. When in doubt – substitute (if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)

So, try, say,  "Which is greater, 5²+3² or (5+3)²?"
In the first case, 5² = 25 and 3² = 9, so it would be 25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first in the order of operations) and get 8. Then you would square that, and get 64, which is clearly greater than 34.
Therefore  (5+3)²  is greater than 5²+3².

For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x²+y² or (x+y)²
= 1+1 or 2 ²
= 2 or 4

What if the example had been a bit different, though? What if it had been:
"Which is greater, x²*y² or (x*y)² (using multiplication instead of addition)?

Can you figure that one out? Try using substitution to figure out the rule and why it works.

Caveat - There is no answer that will work every time. You will see that if you substitute the number 1, the answer will be neither is greater. That's why you shouldn't find this question phrased as an either/or question on a test.

Then, you will always be able to figure this out:
Which is greater, x²*y² or x²+y²? (Same caveat goes for this one as above).

Comment from astute reader Randall Jones:

For the first problem, you can also can solve the problem without substitution.
(x+y)^2= x^2 + 2xy + y^2 which is obviously greater than x^2 + y^2, that is if x and y are both greater than zero.

But you say to substitute any real numbers and if either x or y is negative, than the reverse would be true, x^2 + y^2 would be the larger algebraic expression.

All of the above makes me consider how ineffective and trivial standardized testing is.

You'll never see concepts like these adequately tested. You'll also seldome see these adequately taught, or explored.

Part of the thrill of math is examining and pondering questions like these. It's not about "which is the correct answer all of the time." It's more about looking and seeing and playing with interesting concepts, so that some day when you are faced with similar problems, you can use your experience of pondering these to help you with those.

You can parlay your experience here into "real life," too. When faced with "either/or" questions, you don't have to automatically assume that they deserve either/or answers in all cases. Maybe they do, maybe they don't. You have to look at them closely. Maybe sometimes the question will be dependent on the variables.

That can lead you into thinking about how laws are made, how bureaucracies work (or not), why we sometimes have "exceptions" or "mitigating circumstances" in our laws, the wonders of the US Constitution and some of the other countries' Constitutions that were in fluenced by it, the great Enlightenment thinkers who gave us that Constitution, and on and on. At least that where it is leading me now.

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