x^y < y^x?

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madhan_dc: Junior | Next Rank: 30 Posts; Posts: 24; Joined: Mon Jan 24, 2011 1:37 pm

by madhan_dc » Mon Mar 14, 2011 5:48 pm

if x and y are non zero integers is x^y < y^x?

1. x=y^2

2. y >2

the explanation in the OG is not very clear. Can someone please explain. thanks.

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Source: Beat The GMAT — Data Sufficiency | Mon Mar 14, 2011 5:48 pm

Rich@VeritasPrep: GMAT Instructor; Posts: 147; Joined: Tue Aug 25, 2009 7:57 pmFollowed by: 17 members; Location: New York City; Thanked: 76 times; GMAT Score:770

by Rich@VeritasPrep » Mon Mar 14, 2011 7:24 pm

1. x = y^2

Substitute in the prompt:

x^y < y^x ?

(y^2)^y < y ^ (y^2) ?

y^(2y) < y^(y^2) ?

If y=2, then the answer is NO, because the two terms are equal. ( 2^(2*2) = 2^(2^2) )

If y=3, then the answer is YES, because 3^(2*3) < 3^(3^2) or 3^6 < 3^9.

INSUFFICIENT

2. y>2

If y is a positive odd number and x is negative, then x^y will be negative and y^x will be positive, and thus x^y < y^x, leading to a YES.

But if x=2 and y=4, then x^y = y^x, and the answer is NO.

INSUFFICIENT

Together, 1 and 2 tell us that y>2 and x=y^2.

Go back to the expression we derived from Statement 1: y^(2y) < y^(y^2) ?

If y=3, then we have 3^6 < 3^9, which is true.

If y=4, then we have 4^8 < 4^16, which is true.

If y=5, then we have 5^10 < 5^25, which is true.

Notice that as y gets bigger, the distance between the two terms being compared grows larger and larger and larger. Therefore, no matter what value of y we choose, the inequality will always be true.

SUFFICIENT

Answer: C

Rich Zwelling
GMAT Instructor, Veritas Prep

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Night reader: Legendary Member; Posts: 1337; Joined: Sat Dec 27, 2008 6:29 pmFollowed by: 10 members; Thanked: 127 times

by Night reader » Mon Mar 14, 2011 7:55 pm

@Rich, by completing st(1) we may also get to the condition LHS{2y} ~ {y^2}RHS, deduce y>0 and stop here with assurance. You still suggest plugging in the numbers?
further st(2) Is not Sufficient, BUT combined st(1&2) we get y>0 and y>2 Sufficient.

raz1024 wrote:1. x = y^2

Substitute in the prompt:

x^y < y^x ?

(y^2)^y < y ^ (y^2) ?

y^(2y) < y^(y^2) ?

If y=2, then the answer is NO, because the two terms are equal. ( 2^(2*2) = 2^(2^2) )

If y=3, then the answer is YES, because 3^(2*3) < 3^(3^2) or 3^6 < 3^9.

INSUFFICIENT

2. y>2

If y is a positive odd number and x is negative, then x^y will be negative and y^x will be positive, and thus x^y < y^x, leading to a YES.

But if x=2 and y=4, then x^y = y^x, and the answer is NO.

INSUFFICIENT

Together, 1 and 2 tell us that y>2 and x=y^2.

Go back to the expression we derived from Statement 1: y^(2y) < y^(y^2) ?

If y=3, then we have 3^6 < 3^9, which is true.

If y=4, then we have 4^8 < 4^16, which is true.

If y=5, then we have 5^10 < 5^25, which is true.

Notice that as y gets bigger, the distance between the two terms being compared grows larger and larger and larger. Therefore, no matter what value of y we choose, the inequality will always be true.

SUFFICIENT

Answer: C

My knowledge frontiers came to evolve the GMATPill's methods - the credited study means to boost the Verbal competence. I really like their videos, especially for RC, CR and SC. You do check their study methods at https://www.gmatpill.com

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