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

Math questions from mba.com and GMAT Prep software
RonPurewal
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### Re:

MBA Action wrote:I used the Pythagorean theorem and a characteristic of triangles to prove that (1) is not suff. and (2) also.

Can anybody guess how?

well, i can imagine several associations you may have made between the pythagorean theorem, the triangle inequality, and the given statements.

but which one in particular?
sh.bharath
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### Re: Is x^4 + y^4 > z^4?

Ron, You are awesome..
Thanks for this lesson on DS. I had problems in DS and this particular principle is working well for me.

takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.
RonPurewal
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### Re: Is x^4 + y^4 > z^4?

sure.

on the other hand, if you've plugged enough numbers to be quite sure about sufficiency, then QUIT and guess "sufficient" (i.e., don't continue to hunt for proof). remember, time management is king.
abhitechie
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### Re: Is x^4 + y^4 > z^4?

I am amazed this takeaway by Ron is missing from Advanced GMAT Quant.
jnelson0612
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### Re:

MBA Action wrote:I used the Pythagorean theorem and a characteristic of triangles to prove that (1) is not suff. and (2) also.

Can anybody guess how?

Sounds intriguing . . . do tell!
Jamie Nelson
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skullz03
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### Re:

RonPurewal wrote:in my learned opinion, this is one of the most difficult inequality problems that gmatprep has put out there in some time, so you definitely shouldn't feel bad about tanking it.

along the way, we're going to learn 2 VERY important takeaways about data sufficiency number plugging. in fact, the first takeaway is so important that i'll state it 3 times.
here it is for the first time:
takeaway #1: when you plug numbers on a DS problem, YOUR GOAL IS TO PROVE THAT THE STATEMENT IS INSUFFICIENT.
therefore, as soon as you get a 'yes' answer, you should be TRYING to get a 'no' answer to go along with it; and, as soon as you get a 'no' answer, you should be TRYING to get a 'yes' answer to go along with it.

--

statement (2)
you need to pick numbers such that x + y > z, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0.
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x + y > z.
fortunately, this is somewhat simple to do: just make z a big negative number.
try x = 1, y = 1, z = -100
in this case, x + y > z (satisfying statement two), but x^4 + y^4 is clearly less than z^4, so, NO to the prompt question.
insufficient.

--

statement (1)
you need to pick numbers such that x^2 + y^2 > z^2, per this statement.
first, pick a completely random set of numbers that does this: how about x = 1, y = 1, z = 0 (the same set of numbers we picked last time).
these numbers give a YES answer to the prompt question, since 1^4 + 1^4 is indeed greater than 0^4.
now remember: your goal is to prove that the statement is INSUFFICIENT.
this means that we have to try for a 'no' answer.
this means that we have to make z^4 as big as possible, while still obeying the criterion x^2 + y^2 > z^2.
unfortunately, this isn't as easy to do as it was last time; we can't just make z a huge negative number, because z^2 would then still be a giant positive number (thwarting our efforts at obeying the criterion).
so, we have to finesse this one a bit, but the deal is still to make z as big as possible while still obeying the criterion.
let's let x and y randomly be 3 and 3.
then x^2 + y^2 = 18. we need z^2 to be less than this, but still as big as possible. so let's let z = 4 (so that z^2 = 16, which is pretty close).**
with these numbers, x^4 + y^4 = 162, which is much less than z^4 = 256. therefore, NO to the prompt question, so, insufficient.

--

by the way, you may have noticed that divya didn't get the algebra to work, so she just tossed her electronic hands in the air and said 'i give up'.
now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.

since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.

There is no explanation for the statements being used together ? "Option C" in every DS problem ?
tim
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### Re: Is x^4 + y^4 > z^4?

can you please rephrase your post so as to make it clear what you're asking? i honestly can't tell what your question is..
Tim Sanders
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suhail
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### Re: Is x^4 + y^4 > z^4?

RonPurewal wrote:
now, clearly, NOT FIGURING OUT the algebra doesn't PROVE that a statement is insufficient, but, whether intentionally or not, divya is onto something here. specifically:
takeaway #2: if a statement is sufficient, then you WILL be able to PROVE that it is, algebraically or with some other form of theory. in other words, you'll never get a statement that's sufficient, but for which you can only figure that out by number plugging.

since the algebra just doesn't work out - especially for a student as strong as divya (she has posted some pretty amazing stuff on other threads) - you should have a strong inclination to think that the statements are insufficient.
and you'd be right.

Hi Ron,

Is there any good way to decide whether employing algebra OR "plugging in" would be a better way to approach a specific problem? In short, how can we train our minds to look at each problem differently i.e. Is there some way to guide this intuition?

Trying out algebra on this kind of a tough problem wastes at least a minute or so of valuable time, until we are left confused with the algebra not working out. This is not very conclusive either, unless we are confident that we are doing everything possible with the algebra and not just tossing our hands in the "i give up" mode. (I was not a hundred percent sure.)

Looking forward to hearing your thoughts on this.

Regards
RonPurewal
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### Re: Is x^4 + y^4 > z^4?

suhail wrote:Is there any good way to decide whether employing algebra OR "plugging in" would be a better way to approach a specific problem?

yep -- whatever you think of first, you should do.

you should never be in a situation where you're "sitting on" a solution method you've already come up with, because you're trying to think of a "better" method. (that's the surest way to run out of time.)
if something comes to mind ... do it.

Trying out algebra on this kind of a tough problem wastes at least a minute or so of valuable time, until we are left confused with the algebra not working out.

whenever you are doing anything on any problem, you should regularly check yourself with the following 3 questions:
1/ Do I know EXACTLY WHAT I am doing?
2/ Do I know EXACTLY WHY I am doing it?
3/ Am I honestly making any progress?

As soon as these 3 answers are not ALL "yes", you should quit immediately.

if you can do that, you will have no time-management issues at all, and you will have plenty of time to try out all the solution methods you'd ever have to try.

if you are worried about "wasting 1 minute of valuable time", that is what to focus on -- the ability to (a) be brutally honest with yourself about whether you are stuck, and (b) quit.
if you can do those things, you'll have all the "minutes of valuable time" that you could possibly need.
JbhB682
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### Re: Re:

jnelson0612 wrote:
MBA Action wrote:I used the Pythagorean theorem and a characteristic of triangles to prove that (1) is not suff. and (2) also.

Can anybody guess how?

Sounds intriguing . . . do tell!

B -- pretty easy to see that its NS

A -- To prove in sufficiency : The trigger for me is that Pythagoras may somehow be involved

Assume any Pythagoras triplet (3/4/5) and say
---- X^2 : 3
---- Y ^2 : 4
---- Z^2 : 5

So 3+4 = 7 > 5 --- this satisfies S1

now checking on the question stem

The question asks if x^4 + y^4 > z^4

but we know (3)^2 + (4)^2 = (5)^2 per Pythagoras theorem

Hence not sufficient
Sage Pearce-Higgins
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### Re: Is x^4 + y^4 > z^4?

As Ron detailed above, testing cases is definitely the way to go on this problem. Sure, there's a vague connection with Pythagoras, as we've got perfect squares here, but I'm not sure that it's so useful. After all, Pythagoras' theorem is concerned with perfect squares that sum to make perfect squares, and here we've got inequalities.

Your case of (x, y, z) = (3, 4, 5) is good for testing statement 2 (it provides a "no" case), but after that we have to keep looking for a "yes" cases. And, of course, this case isn't any use for statement 1, as it conflicts with statement 1.
YanqingH809
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### Re: Is x^4 + y^4 > z^4?

Sage Pearce-Higgins
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### Re: Is x^4 + y^4 > z^4?

Good to see that!