If a^4b^4, is ab?

Question

Tricky Question.

If  a^4>b^4 , is  a>b  ?

(1)  a>0

(2)  b>0

chetan2u · Answer

Hi..
Not a 700 level Q..

with even powers, the base can be POSITIVE or NEGATIVE 
This is what is to be checked...
If  a^4>b^4 ,  a>b  will depend on whether a is NEGATIVE or POSITIVE...

Let's see the statements..
1) a>0...
Straight sufficient..
Ans is YES a>B

2) B>0..
Nothing about a..
Insufficient

A

BillyZ · Answer

Dear  chetan2u , However, for statement (1), we do not have info about  b  integer.

HKD1710 · Answer

I have answered this question writing the entire thinking process start till end. I made my mind up that i will not write a word for this question. Here is how i did it in my mind counting on the experiences and learning from Quant questions that i ever did before.

it says \(a^4 > b^4\) - so firstly i thought about what does it mean?

it means that both a & b could be negative or positive. i also thought about fractions but that i remember should be thought about when the questions is other way round. say you are given \(a > b\) and asked is \(a^4 > b^4\). isn't it?

so i got pretty sure that i need to know the signs and also it was visible from the statements that it's all about signs of a and b in this question.

i picked stmt-2 first. i generally do it when i see a DS question looking very easy but demands thinking. so i feel better off picking up stmt-2 than stmt-1 first to get away woth trap as much as i can.

(2) \(b > 0\)
what about a? it can be +ve or negative. in any case \(a^4 > b^4\) would be true but when a is -ve then \(a < b\) and vice-versa.
insufficient.

(1) \(a > 0\)
if a is +ve and \(a^4 > b^4\) then does the sign of b matters? NO! why? because if b is -ve then \(a > b\) anyway because a is +ve. BUT if b is +ve, does \(a > b\) still holds? Yes! Why? because it's given that \(a^4 > b^4\) that means the number a is bigger than b.
Sufficient.

Answer is A

BillyZ · Answer

Official solution from Math Revolution.

If you modify the original condition and the question, you get  a^4> b^4 ,  a^4-b^4>0 ,  (a^2-b^2)(a^2+b^2)>0 , and  a^2+b^2>0  is always established, so you get  a^2-b^2>0  and  (a-b)(a+b)>0 . If you modify the question, you get  a>b ? or  a-b>0 ? In order to get  (a-b)(a+b)>0 , you need to get  a+b>0 ? So it becomes  a-b>0 ? and  a+b>0 ? and if you add the two questions,  b  gets deleted and from  2a>0 ?, you get  a>0 ? Hence, (1) is yes and sufficient.

For (2)  a=2  and  b=1  yes, but if it is  a=-2  and  b=1 , it is no, hence it is not sufficient. Therefore, the answer is  A .

Dear  HKD1710 , Does it sound better? The other way round of solving this question.

Bunuel · Answer

There is an easier way.

If  a^4>b^4 , is  a>b  ?

Taking the 4th root from  a^4>b^4  gives |a| > |b|. So, it's given that a is further from 0 than b is. Now, a will be greater than b if a is positive. So, basically if we have one of the following cases:

----b--0---------a----
-------0--b------a----

(1)  a>0 . Sufficient.

(2)  b>0 . Not sufficient.

Answer: A.

BillyZ · Answer

Dear  Bunuel , What if statement (1) changed to  a<0  ? Is  |a| > |b|  ?

----b----a----0----

Bunuel · Answer

What is your question? What would be the answer if (1) were a < 0 instead of a > 0?

If yes, then the answer would remain the same.

a < 0 means that we have one of the following cases:

----a----b--0---------
----a-------0--b------

In any case a < b.

chetan2u · Answer

Hi...
Refer your Q earlier too..
This Q mainly hinges on sign of a...

If a <0... ans to a>b will be NO and again statement will be sufficient..

For a^4 >b^4, numeric value of a has to be greater than numeric value of b.. or |a|>|b|
So if a is NEGATIVE, it will be further from 0 as compared to B..
Irrespective of SIGN of b, a will always be LESS than b..

Businessconquerer · Answer

Can someone clear me on this?

We can take values as +, -, Decimals, zero
As we haven't been told in the question, that a,b are integers, you know. I would have made that mistake to be frank
So taking statement 1 for example
If a>0,
Integer wise
3^4 > 2^4 : a>b
Decimal wise
(0.3)^4 > (0.4)^4 : a<b

How is the answer A????
 Similarly for the other side
Even if you have both >0 you still won't get an answer

I believe that E must be the answer

Bunuel   GMATNinja   EMPOWERgmatRichC   carcass   egmat   VeritasKarishma 
Anyone?

KarishmaB · Answer

In this question, integers/decimals are irrelevant.

Given:  a^4>b^4 
What does this imply? It implies |a| > |b|

So a and b could be negative/positive but in any case, the absolute value of a is greater than the absolute value of b.

Question: Is a>b? 
Depends on the signs of a and b.

(1)  a>0

a is positive and its absolute value is more than that of b.
b can be positive or negative. 
So say a = 3, b = 2
or a = 3, b = -2
In any case, a will be greater than b. Sufficient

(2)  b>0

b is positive. a could be negative or positive. 
For example:
a= 3, b = 2
a = -3, b = 2
So a may or may not be greater than b. Not sufficient

Answer (A)

Kinshook · Answer

Asked: If a^4>b^4 , is a>b ? |a|>|b| (1) a>0 Since a>0 & |a|>|b| a>b regardless of sign of b SUFFICIENT (2) b>0 Since if a<0; a0; a>b NOT SUFFICIENT IMO A

Bunuel · Answer

The red part above is not true. (0.3^4 = 0.0081) < (0.4^4 = 0.0256). The reason the answer is A is given in several solutions above. For example: https://gmatclub.com/forum/if-a-4-b-4-i ... l#p1807575 Hope it helps.

Businessconquerer · Answer

tHat actually does help

I was under the impression that between 0 and 1 the greater the power the smaller the value what I didn't consider is that the base has to be the same for it
Anyhow lesson learnt
Ultimately now I know
Also thANKs to  VeritasKarishma  I read your solution too. It's just that on test day if the absolute value trick doesn't strike, I will have to go old school.

Bunuel · Answer

Yes, if 0 < x < 1, then x > x^2 > x^3 > ... But this does not mean that if x < y (where 0 < x < 1 and 0 < y < 1), then x^2 > y^2. If x < y (where 0 < x < 1 and 0 < y < 1), then x^n < y^n (where n is a positive integer).

EMPOWERgmatRichC · Answer

Hi Businessconquerer,

We're told that A^4 > B^4. We're asked if A > B? This is a YES/NO question and can be solved with a mix of Number Properties and TESTing VALUES.

To start, we need to consider the initial information that A^4 is greater than B^4. Since we're dealing with EVEN Exponents, the end result of each of those calculations will either be POSITIVE or ZERO. For example:

(2)^4 = 16
(-3)^4 = 81
(0)^4 = 0

Thus, you could end up with a situation in which A > B (for example, A = 3, B = 2) OR B > A (for example, A = -3, B = 2).

1) A > 0

Since A is positive, we end up with an "upper limit" for what B can equal. For example...

IF.... A = 3, then A^4 = (3)^4 = 81

Regardless of whether B is positive, negative or zero, the value of B^4 MUST be less than 81. That 'limitation' means that B must fall into the range: -3 < B < 3. No matter which of those values we consider, the value of A (in this case, "3") will ALWAYS be greater than the value of B, so the answer to the question is YES.

This exact situation occurs with every potential positive value of A that we could choose. B might also be positive (but it would have to be LESS than A.... or B could be zero or a negative (and in those situations, B would also clearly be LESS than A), so the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT.

2) B > 0

IF.... B = 1, then B^4 = (1)^4 = 1

IF.... A = 2, then the answer to the question is YES.
IF.... A = -2, then the answer to the question is NO.
Fact 1 is INSUFFICIENT.

Final Answer:

Show Spoiler

A

GMAT assassins aren't born, they're made,
Rich

KarishmaB · Answer

Note that the point about absolute values is a shortcoming of our schooling system, not a trick. To make things "easier" for schools, we teach them with many constraints (say, the numbers are positive) and that is why we grow up not completely understanding the concepts. Nevertheless, try to put in effort to understand them now - you will find it useful not only during the test but also in your MBA coursework.

The moment you see a^2 = b^2 (or some other even power), it should make you think that this implies |a| = |b|.
a and b may not actually be equal since they may have different signs but their absolute values surely are. 
The same thing will hold for inequalities too.

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If a^4>b^4, is a>b?

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