If u > 0 and v > 0, which is greater

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jjjinapinch: Senior | Next Rank: 100 Posts; Posts: 83; Joined: Mon Jul 24, 2017 8:16 amFollowed by: 1 member

by jjjinapinch » Mon Jul 24, 2017 9:28 am

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If u > 0 and v > 0, which is greater, u^v or v^u?
(1) u = 1
(2) v > 2

Official Guide question
Answer: C

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Source: Beat The GMAT — Data Sufficiency | Mon Jul 24, 2017 9:28 am

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Mon Jul 24, 2017 5:07 pm

Using Statement 1, if u = 1, then

u^v = 1^v = 1

and

v^u = v^1 = v

So the question "which is greater, u^v or v^u?" becomes "which is greater, 1 or v?" We don't know if v is greater than 1 (we only know that v > 0) so Statement 1 is not sufficient. But when we use both statements, we know v > 2, so v is certainly greater than 1, and the answer is C.

Statement 2 is not sufficient alone. We just saw above that v^u can be larger, but using only statement 2, it might be that u = 3 and v = 3, say, and then neither of u^v or v^u is larger than the other, so we can get two different answers to the question only using Statement 2. Other sets of values will prove that as well - I think the OG solution uses u = 3 and v = 4.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by [email protected] » Mon Jul 24, 2017 5:27 pm

Hi jjjinapinch,

This question can be solved by TESTing VALUES.

We're told that both U and V are POSITIVE (and while that doesn't necessarily mean that they have to be positive integers, we can use integers to get to the correct answer). We're asked whether U^V or V^U is greater. This is the equivalent of a YES/NO question.

1) U = 1

IF.....
V = 1
then 1^1 and 1^1 are the SAME value, so NEITHER is greater.

IF....
V = 2
then 1^2 = 1 and 2^1 = 2, so V^U is greater.
Fact 1 is INSUFFICIENT

2) V > 2

IF....
V = 3, U = 1
then 1^3 = 1 and 3^1 = 3, so V^U is greater.

IF....
V = 3, U = 3
then 3^3 and 3^3 are the SAME value, so NEITHER is greater.
Fact 1 is INSUFFICIENT

Combined, we know....
U = 1 and V > 2
U^V will always be 1^(something > 2) = 1 and V^U = (something greater than 2)^1 = that same value greater than 2. Thus, V^U will ALWAYS be greater.
Combined, SUFFICIENT

Final Answer:
C

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Rich

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8142; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri Dec 08, 2017 6:55 am

jjjinapinch wrote:If u > 0 and v > 0, which is greater, u^v or v^u?
(1) u = 1
(2) v > 2

We are given that u > 0 and that v > 0. We must determine whether u^v is greater than v^u.

Statement One Alone:

u = 1

If u = 1, then u^v = 1^v = 1 (recall that 1 raised to any power is 1) and v^u = v^1 = v (recall that any number raised to the 1st power is itself). However, since we do not know the value of v, we cannot determine whether u^v is greater than v^u.

For example, if v = 2, then u^v < v^u (1^2 < 2^1). However, if v = Â½, then u^v > v^u because 1^(1/2) > (1/2)^1. Statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

v > 2

Although we know that v > 2, since we do not have any information about u, we cannot answer the question. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two, we know that u = 1 and that v > 2. Thus, we see that:

u^v = 1^v = 1 and v^u > 2^1 (since v > 2). Since v^u > 2 and u^v = 1, v^u is greater than u^v.

Answer: C

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