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If a and b are nonzero integers, which of the following must be

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kingflo
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If a and b are nonzero integers, which of the following must be [#permalink] New post   Updated on: 20 Jul 2013, 10:37
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If a and b are nonzero integers, which of the following must be negative?

A. \((-a)^{-2b}\)

B. \((-a)^{-3b}\)

C. \(-(a^{-2b})\)

D. \(-(a^{-3b})\)

E. None of these
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C

Originally posted by kingflo on 20 Jul 2013, 10:12.
Last edited by kingflo on 20 Jul 2013, 10:37, edited 2 times in total.
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Bunuel
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   20 Jul 2013, 10:34
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If a and b are nonzero integers, which of the following must be negative?

A. \((-a)^{-2b}=\frac{1}{(-a)^{2b}}=positive\).

B. \((-a)^{-3b}\) --> may be positive (consider a=1 and b=2) as well as negative (consider a=1 and b=1).

C. \(-(a^{-2b})=-\frac{1}{a^{2b}}=-\frac{1}{positive}=negative\)

D. \(-(a^{-3b})\) --> may be positive (consider a=-1 and b=1) as well as negative (consider a=1 and b=1).

E. None of these

Answer: C.
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   17 Jan 2018, 16:33
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kingflo wrote:
If a and b are nonzero integers, which of the following must be negative?

A. \((-a)^{-2b}\)
B. \((-a)^{-3b}\)
C. \(-(a^{-2b})\)
D. \(-(a^{-3b})\)
E. None of these



IMPORTANT CONCEPT
Rule #1: EVEN powers are always greater than or equal to zero.
So, (POSITIVE value)^(EVEN integer) > 0, and (NEGATIVE value)^(EVEN integer) > 0


So, the correct answer here is C. Here's why:


C) –[a^(-2b)]
Since b is an integer, we know that -2b is an EVEN integer.
So, we get: –[a^(EVEN integer)]
By our rule, a^(EVEN integer) is greater than or equal to zero
Since a is a NON-zero integer, we can conclude that a^(EVEN integer) is GREATER THAN zero
In other words, a^(EVEN integer) is POSITIVE
This means that -[a^(EVEN integer)] is NEGATIVE

Answer:
Show Spoiler
C


Cheers,
Brent
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gagan543
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   16 Apr 2020, 05:27
i have a confusion in option D it is already given a and b is a nonnegative number
so using same method to solve option D as you used for solving option C
D would also be always negative Bunuel
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Bunuel
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   16 Apr 2020, 05:35
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gagan543 wrote:
Bunuel wrote:
If a and b are nonzero integers, which of the following must be negative?

A. \((-a)^{-2b}=\frac{1}{(-a)^{2b}}=positive\).

B. \((-a)^{-3b}\) --> may be positive (consider a=1 and b=2) as well as negative (consider a=1 and b=1).

C. \(-(a^{-2b})=-\frac{1}{a^{2b}}=-\frac{1}{positive}=negative\)

D. \(-(a^{-3b})\) --> may be positive (consider a=-1 and b=1) as well as negative (consider a=1 and b=1).

E. None of these

Answer: C.



i have a confusion in option D it is already given a and b is a nonnegative number
so using same method to solve option D as you used for solving option C
D would also be always negative Bunuel


\(-(a^{-3b})=-\frac{1}{a^{3b}}\).

\(a^{3b}=a^{odd}\) can be negative as well as positive, while \(a^{2b}=a^{even}\) is always positive.
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gagan543
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   16 Apr 2020, 05:48
i still don't understand, a^3b will always be positive because the question says a & b are not negative so ,so assuming a=1 b=1
a^3(1) will become positive and -(1/positive) will become negative as we cannot take take a=-1 or b=-1
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Bunuel
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   16 Apr 2020, 05:50
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gagan543 wrote:
i still don't understand, a^3b will always be positive because the question says a & b are not negative so ,so assuming a=1 b=1
a^3(1) will become positive and -(1/positive) will become negative as we cannot take take a=-1 or b=-1


No, the question says "If a and b are nonzero integers, ", NOT non-negative.
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gagan543
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   16 Apr 2020, 05:55
oh ,thanks i miss read it
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Re: If a and b are nonzero integers, which of the following must be [#permalink] New post   25 Dec 2023, 00:51
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Re: If a and b are nonzero integers, which of the following must be [#permalink]
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