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# M13-32

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Math Expert
Joined: 02 Sep 2009
Posts: 58473

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16 Sep 2014, 00:50
1
2
00:00

Difficulty:

85% (hard)

Question Stats:

41% (01:42) correct 59% (02:02) wrong based on 68 sessions

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Is $$a^b \gt b^a$$ ?

(1) $$a \gt b \gt 1$$

(2) $$a = b + 1$$

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Math Expert
Joined: 02 Sep 2009
Posts: 58473

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16 Sep 2014, 00:50
Official Solution:

Statements (1) and (2) combined are insufficient. Consider:

$$a = 3$$, $$b = 2$$. $$a^b = 3^2 = 9 \gt b^a = 2^3 = 8$$. The answer is "yes".

$$a = 4$$, $$b = 3$$. $$a^b = 4^3 = 64 \lt b^a = 3^4 = 81$$. The answer is "no".

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Intern
Joined: 01 Feb 2017
Posts: 3
Location: India
GMAT 1: 690 Q47 V37
GPA: 4

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08 Apr 2018, 22:26
For values less than zero even both the statements are not enough.
check with values less than zero
Intern
Joined: 02 Feb 2018
Posts: 31

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02 Nov 2018, 10:30
You can save a lot of time on this one when you start by picking numbers that satisfy both statements at the same time as Bunuel did.. I wonder if this should be the default state when facing DS questions where picking numbers is obviously the best way to go - any advice on this? Or what specific signaled you in this question that this is the best strategy?
VP
Joined: 14 Feb 2017
Posts: 1220
Location: Australia
Concentration: Technology, Strategy
Schools: LBS '22
GMAT 1: 560 Q41 V26
GMAT 2: 550 Q43 V23
GMAT 3: 650 Q47 V33
GMAT 4: 650 Q44 V36
WE: Management Consulting (Consulting)

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14 May 2019, 22:48
Stat.1
3> 2> 1
a=2, b=3

3^2 > 2^3?
9 is not greater than 8 -->No

Test for Yes
4>2>1
a=4, b=2
4^2 > 2^4?
16>16? No

Therefore Stat.(1)--> Insufficient

Statement 2
a= b+1
-1=-2+1
a=-1 b = -2
1>-1/2 ? Yes

a= 4, b= 3
4=3+1
4^3 > 3^4?
64>81? No
Statement (2) --> Insufficient

1+2 combined we have b+1>b>1

a and b are consecutive. One case 3>2>1 is already valid for Yes ( that a^b > b^a)

So lets test for no:

5>4>1

5^4 > 4^5?
625 > 1024? No.

Therefore E
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Manager
Joined: 15 Jul 2016
Posts: 101
Location: India
Schools: Oxford "21 (A)
GMAT 1: 690 Q48 V36

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22 Jul 2019, 23:30
1
It's good to remember the following rule...

IF there are 2 positive numbers, then the number which has a greater power will be greater.
eg; 8^9 > 9^8 even though 9>8.

However, there are two exceptions to the rule:

1) 2^3 < 3^2
==> 8<9

2) 2^4 = 4^2
==> 16 = 16.

Once you know these properties, it'll be easier two pick the numbers. As you can see, the 1st exception satisfies both the conditions.

Thanks!
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Please give Kudos if you agree with my approach :)
Manager
Joined: 03 Sep 2018
Posts: 179

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13 Oct 2019, 22:43
578vishnu wrote:
It's good to remember the following rule...

IF there are 2 positive numbers, then the number which has a greater power will be greater.
eg; 8^9 > 9^8 even though 9>8.

However, there are two exceptions to the rule:

1) 2^3 < 3^2
==> 8<9

2) 2^4 = 4^2
==> 16 = 16.

Once you know these properties, it'll be easier two pick the numbers. As you can see, the 1st exception satisfies both the conditions.

Thanks!

Bunuel could you confirm this?
Re: M13-32   [#permalink] 13 Oct 2019, 22:43
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# M13-32

Moderators: chetan2u, Bunuel