Is x^4 + y^4 > z^4?

by **abramson** Mon Sep 03, 2007 6:01 pm

Hello folks,

I'm taking the test in less than 48 hours, so any help will be appreciated. I don't know how to go about this problem.

Thanks!

by **abramson** Mon Sep 03, 2007 6:02 pm

I guess plugging in numbers could eventually give the answer, but is there any quick method to test this?

Thanks!

by **abramson** Mon Sep 03, 2007 6:05 pm

official answer is (E)

by **StaceyKoprince** Sun Sep 09, 2007 4:20 pm

Hi, abramson - I'm sorry we didn't get to this for you before your test. We've been a few days behind lately so it's likely that no one saw this within the 48 hour window and then instructors who saw it after didn't answer b/c they thought it was too late. But I wanted to reply just in case you're still wondering.

Plugging in numbers is actually very efficient here if you're using the right process. First, make sure that you think of trying positives, negatives, fractions, zero, etc - the key is to be smart about which numbers you choose based upon the given problem. Also, any time you try numbers on a yes / no DS, you want to try to disprove the statement - if you find a yes, look for a no, and vice versa.

So, is x^4 + y^4 > z^4? yes/no

(1) x^2 + y^2 > z^2
x = 1, y = 2, z = 0. These three numbers make statement 1 true, so they are valid choices. Then, x^4 + y^4 > z^4 is also true, so I can answer the question "yes." Now, since I've found a "yes" I want to see if I can find a "no." I know that, normally, raising something to an even exponent makes it a bigger number - but that's not true if the starting numbers are fractions between zero and one. Then, the numbers get smaller. So I want to try that next.
x = 0.5, y = 0.9, z = 1. These three numbers make statement 1 true, so they are valid choices. Then, x^4 + y^4 > z^4 is NOT true, so I can answer the question "no."
Yes + no = insufficient.

(2) x + y > z
Because I'm doing the second statement now, I want to see if I can reuse any work I already did for the first statement - that will make my job easier.
Can I use x = 1, y = 2, z = 0? These three numbers make statement 2 true, so they are valid choices. Then, I already know that x^4 + y^4 > z^4 is also true, so I can answer the question "yes" without having to do any more calculations.
Can I use x = 0.5, y = 0.9, z = 1? These three numbers make statement 2 true, so they are valid choices. Then, I once again already know that x^4 + y^4 > z^4 is NOT true, so I can answer the question "no" without having to do any more calculations.
Yes + no = insufficient.

(1) AND (2)
Here's the real value of checking to see whether I could use the same numbers for statement 2: the work to assess the two statements together is already done. I know I can get a yes and a no looking at them together b/c I used the same numbers for the two scenarios up above.
Yes + no = insufficient.

Answer E.