In the context of this question, there's no way to get -1/4. Clearly, no positive number what yield this result. If you test negatives (even without considering the statements) you'll get the following
a = -1; a^a = (-1)^-1 = -1
a = -2; a^a = (-2)^(-2) = 1/4
a = -3; a^a = (-3)^(-3) = -1/27
The numbers will alternate + and -, and will become smaller and smaller in absolute magnitude.
a is a nonzero integer
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DavidG@VeritasPrep
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Brent@GMATPrepNow
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Asma77 wrote:a is a nonzero integer. Is a^a greater than 1?
(1) a < -1
(2) a is even
Target question: Is a^a greater than 1?
Given: a is a nonzero integer.
Statement 1: a < -1
Let's start TESTING some values of a and see if we discover any PATTERNS
a = -2. Here a^a = (-2)^(-2) = 1/(-2)^2 = 1/4
a = -3. Here a^a = (-3)^(-3) = 1/(-3)^3 = 1/(-27) = - 1/27
a = -4. Here a^a = (-4)^(-4) = 1/(-4)^4 = 1/256
a = -5. Here a^a = (-5)^(-5) = 1/(-5)^5 = 1/(some negative value) = something negative
a = -6. Here a^a = (-6)^(-6) = 1/(-6)^6 = 1/(some positive integer) = a positive fraction that's less than 1
a = -7. Here a^a = (-7)^(-7) = 1/(-7)^7 = 1/(some negative value) = something negative
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.
.
As we can see, when a is a NEGATIVE EVEN number, a^a = some positive fraction that's LESS THAN 1
When a is a NEGATIVE ODD number, a^a = some negative value that's LESS THAN 1
So, in both possible cases, a^a is definitely LESS THAN 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: a is even
We already saw in statement 1 that if a is a NEGATIVE EVEN integer, then a^a = some positive fraction that's LESS THAN 1
What if a is a POSITIVE EVEN integer?
Let's test some possible values:
a = 2. Here a^a = (2)^(2) = 4. Here, a^a is GREATER THAN 1
So, we'll stop right here.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Answer = A
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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towerSpider
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a. if a = 1: you know answer -> Answer is No; No, a^a (1*1) is NOT greater than 1Asma77 wrote:a is a nonzero integer. Is a^a greater than 1?
(1) a < -1
(2) a is even
I have question here if (-2)^(-2) equal to 1/4 so how we can write -1/4 as the expression a^a
b. if a < 1: you know answer -> a^a would be in form 1/no. and therefore less than 1.
c. if a > 1: you know answer. -> this would make a^a greater than 1.
(1) this is part of b. and you would know the answer.
(2) Don't know. Different answers for 2 and -2.
Answer is A
People are not prisoners of fate, but prisoners of their own mind.
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Matt@VeritasPrep
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Let's start with aᵃ > 1.Nina1987 wrote:Is there an algebraic way to solve this question? thanks
Glossing over a lot of complexity ...
If a is negative, then aᵃ is either imaginary or less than 1.
If a is 0, then we get an indeterminate solution.
If 1 > a > 0, then aᵃ < 1.
If a = 1, then aᵃ = 1.
Our only case left is a > 1, which of course works, so aᵃ > 1 is equivalent to a > 1.
From here, the statements are pretty easy to work with.
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Nina1987
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Matt, I find that solution poetic. Thanks!
Matt, one more question- Who is a typical Q51 test taker?
-One who is eloquent in such approach?
or
-One who may not be really versed in such approaches but incredibly adept at number picking?
Matt, one more question- Who is a typical Q51 test taker?
-One who is eloquent in such approach?
or
-One who may not be really versed in such approaches but incredibly adept at number picking?
Matt@VeritasPrep wrote:Let's start with aᵃ > 1.Nina1987 wrote:Is there an algebraic way to solve this question? thanks
Glossing over a lot of complexity ...
If a is negative, then aᵃ is either imaginary or less than 1.
If a is 0, then we get an indeterminate solution.
If 1 > a > 0, then aᵃ < 1.
If a = 1, then aᵃ = 1.
Our only case left is a > 1, which of course works, so aᵃ > 1 is equivalent to a > 1.
From here, the statements are pretty easy to work with.
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DavidG@VeritasPrep
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I agree - Matt's solution is quite elegant. And in answer to your question: both! The virtue of this site is that participants show that there are a variety of ways to solve any question. The best approach is the one that occurs to you in a reasonable amount of time and leads you to the right answer. If logic similar to Matt's occurs to you in a minute, fantastic! If picking simple #'s gets you to the answer in 90 seconds, great! The important thing is to stay flexible and to remember that if your first approach doesn't seem to be working, there will always be viable alternatives.Matt, I find that solution poetic. Thanks!
Matt, one more question- Who is a typical Q51 test taker?
-One who is eloquent in such approach?
or
-One who may not be really versed in such approaches but incredibly adept at number picking?
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Matt@VeritasPrep
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Hey, thanks! I thought it was kind of boring casework, I wish I had a nicer way of doing it. Graphing was the first thought, but aᵃ isn't an easy graph.Nina1987 wrote:Matt, I find that solution poetic. Thanks!
Matt, one more question- Who is a typical Q51 test taker?
-One who is eloquent in such approach?
or
-One who may not be really versed in such approaches but incredibly adept at number picking?
To be honest, the difference between Q50 and Q51 is attention to detail. One difference between the GMAT and other difficult math tests that work with elementary ideas (such as the AMCs here in the US) is that the GMAT explicitly focuses on questions in which it's easy to make a silly mistake, while the AMC focuses more on questions that are difficult to set up and reward creativity. (More than a decade ago, the AMC eliminated the "silly mistake" style of question.)
The emphasis makes sense, though, because the GMAT is prepping you for a work in which annoying details are very important -- if you miss a detail in a contract or a stock-pricing algorithm, you're screwed! -- while the AMC is testing your mathematical creativity. But I'm more an AMC guy; I tend to make sloppy mistakes fairly often because of boredom or complacency or excitement or whatever, and only realize them about 20 seconds after I've clicked and confirmed my answer!
