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

anyone? Anyone? ANYONE?? :roll:

Ron Purewal has given a great explanation of this question in the following link:

https://www.manhattangmat.com/forums/is- ... t4655.html

In case people are shy about clicking the link:

" Ron Purewal:

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.

answer = e.

--

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."

Ian Stewart wrote:

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

(1) x^2 + y^2 > z^2

(2) x+y > z

I was reminded of this question in a PM. As Ron says above, this is quite a high-level inequalities question.

You might think, before testing numbers, where you may have seen inequalities or equations like those in Statements 1 and 2. Statement 1 looks suspiciously like the Pythagorean Theorem, for example, while Statement 2 looks like the Triangle Inequality (the sum of two sides of a triangle always exceeds the third side). If you see how the inequalities relate to geometry, you can find numbers quite quickly to show that the answer to the question can be 'no', even using both statements:

For example, let:

x^2 = 3 (i.e. let x = root(3))
y^2 = 4 (i.e. let y = 2)
z^2 = 5 (i.e. let z = root(5))

Then x^2 + y^2 > z^2, so S1 is true. Since root(3) + 2 is larger than root(5), S2 is also true. And with these numbers, x^4 + y^4 = 9 + 16 = 25 = z^4, so with these numbers, the answer to the question is 'no'; x^4 + y^4 is not larger than z^4. Since it's clear if we make x and y large and z zero, the answer to the question can also be 'yes', the two statements together are insufficient.

Terrific..Thanks Ian

rabab wrote:I searched through the site. But none of the answers seemed convincing to me. Anyone wants to shed a light on this problem?

OA- E

Under I since there is no mention of integers, assume all real numbers.

Let x and y =1. Then ( x^2 +y^2)= 2. So 1 says that 2> z^2, which means (Z-sqr(2)) (Z+sqr(2)) <0.

The solution to this inequality is (-sqr(2), 0) Union (0, sqr(2)). Z can take any values between these intervals excluding –sqr(2), sqr(2) and 0.

(2^1/4) is one of these numbers. clearly under a
2> 2^1/4
Go back to stem and let x = 1 y =1 and z= (2^1/4)


Stem: 2> Z4
2> ((2^1/4)^4)
2>2 cannot be true

But if Z= 1 which is also in the above intervals under a
2>1, so I cannot answer the question.


Under II

Let Z = -2^1/4, which satisfies condition b if X= 1 y=1. The sume of 2 positive numbers is greater than any negative number.

Stem: 2>(-2^1/4)^4 which cannot be true
Letting Z=1
2>1 which works this time. So II is not sufficient

Moral: If GMAT gives you powers of 4, think root of 4.

Ian Stewart wrote:

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

(1) x^2 + y^2 > z^2

(2) x+y > z

I was reminded of this question in a PM. As Ron says above, this is quite a high-level inequalities question.

You might think, before testing numbers, where you may have seen inequalities or equations like those in Statements 1 and 2. Statement 1 looks suspiciously like the Pythagorean Theorem, for example, while Statement 2 looks like the Triangle Inequality (the sum of two sides of a triangle always exceeds the third side). If you see how the inequalities relate to geometry, you can find numbers quite quickly to show that the answer to the question can be 'no', even using both statements:

For example, let:

x^2 = 3 (i.e. let x = root(3))
y^2 = 4 (i.e. let y = 2)
z^2 = 5 (i.e. let z = root(5))

Then x^2 + y^2 > z^2, so S1 is true. Since root(3) + 2 is larger than root(5), S2 is also true. And with these numbers, x^4 + y^4 = 9 + 16 = 25 = z^4, so with these numbers, the answer to the question is 'no'; x^4 + y^4 is not larger than z^4. Since it's clear if we make x and y large and z zero, the answer to the question can also be 'yes', the two statements together are insufficient.

Very nice approach Ian..

Statement 1:
x^2 + y^2 > 0
z^2 > 0

x^2 + y^2 > z^2
SQUARE ON BOTH SIDES

x^4 + y^4 + 2 * x^2 * y^2 > z^4

2 * x^2 * y^2 --- this can have any value to make RHS = or > than LHS

x^4 + y^4 >, = OR z^4

Insufficient

Statement 2:

x + y > z
SQUARE BOTH SIDES

x^2 + y^2 + 2xy > z^2
2xy takes any value to make LHS = RHS or LHS < or > RHS

x^2 + y^2 >, = OR z^2

and hence x^4 + y^4 >, =, OR z^4

Insufficient

Statement 1 and 2:

Insufficient

Ans E


HOPE THIS HELPS !!!!!!