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If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2

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If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 27 Oct 2005, 15:28
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Question Stats:

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If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2
(2) y > 2

Data Sufficiency
Question: 121
Category: Arithmetic; Algebra Arithmetic operations; Inequalities
Page: 161
Difficulty: 650


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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 04 Aug 2010, 05:52
14
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jpr200012 wrote:
If x and y are nonzero integers, is \(x^y < y^x\)?

(1) \(x = y^2\)
(2) \(y > 2\)


If x and y are nonzero integers, is \(x^y < y^x\)?

(1) \(x = y^2\) --> if \(x=y=1\), then \(x^y=1=y^x\), so the answer would be NO BUT if \(y=3\) and \(x=9\), then \(x^y=9^3<y^x=3^9\), so the answer would be YES. Not sufficient.

(2) \(y>2\). No info about \(x\), not sufficient.

(1)+(2) From (1) \(x = y^2\), thus the question becomes: is \((y^2)^y<y^{(y^2)}\)? --> is \(y^{2y}<y^{(y^2)}\)? Now, since from (2) \(y=integer>2\), then \(2y\) will always be less than \(y^2\), therefore \(y^{2y}\) will be less than \(y^{(y^2)}\). Sufficient.

Answer: C.


jpr200012 wrote:
This is a tricky question. I think it relies on you misapplying the rule: \((x^a)^b = x^{ab}\). Is this only valid if a and b are constants?

Example:
(1) \(x = y^2\);

\(x^y < y^x\) therefore, \((y^2)^y < y^{y^2}\). How do you simplify this? The guide shows to \(y^{2y} < y^{y^2}\). The left hand side makes sense to me.

Why would \(y^{y^2}\) not simplify to \(y^{2y}\) also? Plugging in numbers, it makes sense. I just want to understand the concept.


If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\), which on the other hand equals to \(a^{mn}\).

So:
\((a^m)^n=a^{mn}\);

\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\).

Hope it's clear.
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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 04 Aug 2010, 05:21
This is a tricky question. I think it relies on you misapplying the rule: \((x^a)^b = x^{ab}\). Is this only valid if a and b are constants?

Example:
(1) \(x = y^2\);

\(x^y < y^x\) therefore, \((y^2)^y < y^{y^2}\). How do you simplify this? The guide shows to \(y^{2y} < y^{y^2}\). The left hand side makes sense to me.

Why would \(y^{y^2}\) not simplify to \(y^{2y}\) also? Plugging in numbers, it makes sense. I just want to understand the concept.
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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 05 Aug 2010, 01:25
Posting this msg here even though i sent a private msg to you-for the benefit of others here.

Hi Bunuel, apprecite ur wonderful explanation. I am having trouble in DS question where x & y are termed as non-zero integers.

What is the best way to analyze instances where x & y are are NEGATIVE integers. I see that u have not analyzed this possibility. is there a trick to be sure that this is not needed as u have solved in this case?

Please enlighten. Thanks.
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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 05 Aug 2010, 02:12
1
ramanankris wrote:
Posting this msg here even though i sent a private msg to you-for the benefit of others here.

Hi Bunuel, apprecite ur wonderful explanation. I am having trouble in DS question where x & y are termed as non-zero integers.

What is the best way to analyze instances where x & y are are NEGATIVE integers. I see that u have not analyzed this possibility. is there a trick to be sure that this is not needed as u have solved in this case?

Please enlighten. Thanks.


On DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get a YES answer with one chosen number(s) and a NO with another.

For statement (1) I got YES answer and then NO answer with positive numbers, so my goal to prove that this statement was not sufficient was reached, hence there was no need to try negative numbers.

Hope it's clear.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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New post 23 Feb 2017, 04:54
Prompt analysis
Let the number of tapes with carmen and rafael be x and y respectively.

x+12 = 2y

Superset
Value of x and y will be a whole number.

Translation
St 1: y =x +5. Solving 2 equations we get x = 2, y =7. ANSWER
St 2: x<12. Cannot determine the exact value of x. INSUFFICIENT

Option A
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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2  [#permalink]

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Re: If x and y are nonzero integers, is x^y < y^x ? (1) x = y^2 (2) y > 2 &nbs [#permalink] 10 Jul 2018, 05:18
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