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

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If a and b are nonzero integers, which of the following must be  [#permalink]

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Updated on: 20 Jul 2013, 09:37
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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

Originally posted by kingflo on 20 Jul 2013, 09:12.
Last edited by kingflo on 20 Jul 2013, 09:37, edited 2 times in total.
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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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20 Jul 2013, 09:34
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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

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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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17 Jan 2018, 15:33
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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

Cheers,
Brent
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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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16 Apr 2020, 04:27
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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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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16 Apr 2020, 04:35
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

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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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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16 Apr 2020, 04:48
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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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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16 Apr 2020, 04:50
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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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Re: If a and b are nonzero integers, which of the following must be  [#permalink]

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16 Apr 2020, 04:55
oh ,thanks i miss read it
Re: If a and b are nonzero integers, which of the following must be   [#permalink] 16 Apr 2020, 04:55