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Math Expert V
Joined: 02 Sep 2009
Posts: 60480
If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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17 00:00

Difficulty:   5% (low)

Question Stats: 85% (01:07) correct 15% (01:30) wrong based on 1237 sessions

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If u > 0 and v > 0, which is greater, $$u^v$$ or $$v^u$$?

(1) u = 1
(2) v > 2

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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The original condition has 2 variables, so we need 2 variables as well. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer.
Attachments variable approach's answer probability.jpg [ 219.74 KiB | Viewed 12918 times ]

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Manager  Joined: 21 Sep 2015
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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8
2
If u > 0 and v > 0, which is greater, u^v or v^u?

(1) u = 1

Since we are not told if u and v are different numbers there are two possibilites:
a) u=v= 1 in which case u^v = v^u
b) They are different numbers in which case v^u will always be greater since u^v = 1 (1 raised to anything is 1)

(2) v > 2

It does not give us any information about u. For example u could be 3 and v could be 4 and vice versa

On combining we know u^v will always be 1 . Therefore v^u is greater. SUFFICIENT

Ans = C

Originally posted by rishi02 on 16 Jun 2016, 07:04.
Last edited by rishi02 on 17 Jun 2016, 00:56, edited 1 time in total.
##### General Discussion
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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from statement 1) u=1 so, (1)^v= 1 and v^1. if v=1 then both are equal otherwise v^1 > 1^v. so not sufficient.
2) statement 2. not sufficient. as we know the value of v only.

1) +2) sufficient because this will eliminate the first case when v=1 from first statement.
Manager  Joined: 18 Jan 2010
Posts: 238
Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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1
Bunuel wrote:
If u > 0 and v > 0, which is greater, u^v or v^u?

(1) u = 1
(2) v > 2

Statement 1
when u = 1, u raised to any power is always 1. Also v^u is always v.

If v>1 then v^u > u^v

If v[b]<1, then v^u < u^v [/b]

Clearly not sufficient.

Statement 2

v > 2

say u = 0.5

(3)^0.5 = 1.732
(0.5)^3 = 0.125

v^u > u^v

v=4; u=-1

v^u = 4^(-1) = (1/4)
u^v = (-1)^4 =1

v^u < u^v

Not sufficient.

Combining these two.
v > 2; u = 1
when u = 1, u raised to any power is always 1. Also v^u is always v.

since v is always > 2, and u is always 1. hence

v^u is always greater than u^v
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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1
adiagr According to ques u>0 so it should not be taken as -1

Statement 2 v > 2 say u = 0.5(3)^0.5 = 1.732(0.5)^3 = 0.125v^u > u^v v=4; u=-1v^u = 4^(-1) = (1/4) u^v = (-1)^4 =1 v^u < u^v Not sufficient.
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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Bunuel Can you please post the OA?

Looks tricky. IMO B should be the answer.

Can anyone solve it by picking up the numbers?
Math Expert V
Joined: 02 Sep 2009
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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anurag16 wrote:
Bunuel Can you please post the OA?

Looks tricky. IMO B should be the answer.

Can anyone solve it by picking up the numbers?

The OA is C. Please refer to the solutions above and ask if anything remains unclear.
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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In statement 2:

u=1 and v=3 Then v^u is greater
u=4 and v=3 Then v^u is greater

Please tell a situation when it will not hold true.
Math Expert V
Joined: 02 Sep 2009
Posts: 60480
Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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anurag16 wrote:

In statement 2:

u=1 and v=3 Then v^u is greater
u=4 and v=3 Then v^u is greater

Please tell a situation when it will not hold true.

You can consider the case when u = v. In this case v^u is NOT greater than u^v.
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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1
If u > 0 and v > 0, which is greater, u^v or v^u?

(1) u = 1

Let v = 10........then 1^10 < 10^1

Let v= 1...........then 1^1 = 1^1

Insufficient

(2) v > 2

same example as above:
Let v = 10........then 1^10 < 10^1

Let v= 10...........then 10^10 = 10^10

Insufficient

Combine 1 & 2

V > 2 & u= 1..........So first example applies

Let v = 10........then 1^10 < 10^1

Sufficient

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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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1
i know this says its a <600 level problem, but i could not disagree more. is it just me or does this seem to be at least 600 level?
Manager  S
Joined: 11 Sep 2019
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Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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MathRevolution wrote:
The original condition has 2 variables, so we need 2 variables as well. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer.

What is this? I did not understand it at all. Please elaborate. The method seems tactful but I am not getting the same
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GMAT 1: 600 Q46 V27 Re: If u > 0 and v > 0, which is greater, u^v or v^u?  [#permalink]

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Bunuel wrote:
If u > 0 and v > 0, which is greater, $$u^v$$ or $$v^u$$?

(1) u = 1
(2) v > 2

$$u^v$$ ?? $$v^u$$

Statement 1 : u=1

$$1^v$$ ?? $$v^1$$

Since we don't know anything about v (except it is positive) statement 1 is insufficient.

Statement 1 : v>2. Let's say 3

$$u^3$$ ?? $$3^u$$

Since we don't know anything about u (except it is positive) statement 2 is insufficient.

Combined :

$$1^3$$ < $$3^1$$

Hence Sufficient.

Note: Even if you take value of v as 2.5, the relation will still hold true.
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You are missing on great learning if you don't know what this is: Project SC Butler Re: If u > 0 and v > 0, which is greater, u^v or v^u?   [#permalink] 14 Dec 2019, 09:21
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