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# M26-32

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Math Expert
Joined: 02 Sep 2009
Posts: 42249

Kudos [?]: 132643 [0], given: 12326

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16 Sep 2014, 01:26
Expert's post
7
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Difficulty:

75% (hard)

Question Stats:

33% (00:49) correct 67% (01:15) wrong based on 36 sessions

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If $$x$$ and $$y$$ are negative integers, then what is the value of $$xy$$?

(1) $$x^y=\frac{1}{81}$$

(2) $$y^x=-\frac{1}{64}$$
[Reveal] Spoiler: OA

_________________

Kudos [?]: 132643 [0], given: 12326

Math Expert
Joined: 02 Sep 2009
Posts: 42249

Kudos [?]: 132643 [1], given: 12326

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16 Sep 2014, 01:26
1
KUDOS
Expert's post
Official Solution:

(1) $$x^y=\frac{1}{81}$$. As both $$x$$ and $$y$$ are negative integers then $$x^y=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}$$ hence $$xy=18$$ or $$xy=12$$. Note that as negative integer ($$x$$) in negative integer power ($$y$$) gives positive number ($$\frac{1}{81}$$), then the power must be negative even number. Not sufficient.

(2) $$y^x=-\frac{1}{64}$$. As the result is negative then $$x$$ must be negative odd number, so: $$y^x=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}$$ hence $$xy=12$$ or $$xy=64$$. Not sufficient.

(1)+(2) Only one pair of negative integers $$x$$ and $$y$$ satisfies both statements $$x=-3$$ and $$y=-4$$. Therefore $$xy=12$$. Sufficient.

_________________

Kudos [?]: 132643 [1], given: 12326

Senior Manager
Joined: 04 Oct 2015
Posts: 395

Kudos [?]: 28 [0], given: 235

Location: Viet Nam
Concentration: Finance, Economics
GPA: 3.56

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13 Jul 2017, 07:54
Bunuel wrote:
Official Solution:

(1) $$x^y=\frac{1}{81}$$. As both $$x$$ and $$y$$ are negative integers then $$x^y=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}$$ hence $$xy=18$$ or $$xy=12$$. Note that as negative integer ($$x$$) in negative integer power ($$y$$) gives positive number ($$\frac{1}{81}$$), then the power must be negative even number. Not sufficient.

(2) $$y^x=-\frac{1}{64}$$. As the result is negative then $$x$$ must be negative odd number, so: $$y^x=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}$$ hence $$xy=12$$ or $$xy=64$$. Not sufficient.

(1)+(2) Only one pair of negative integers $$x$$ and $$y$$ satisfies both statements $$x=-3$$ and $$y=-4$$. Therefore $$xy=12$$. Sufficient.

Very good question.
In statement 2, I forget the SPECIAL CASE in which x = -1.
_________________

Do not pray for an easy life, pray for the strength to endure a difficult one - Bruce Lee

Kudos [?]: 28 [0], given: 235

Intern
Joined: 13 May 2017
Posts: 3

Kudos [?]: 0 [0], given: 109

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12 Sep 2017, 06:17
bunuel ,

your questions are amazing and fantastic.

Kudos [?]: 0 [0], given: 109

Re: M26-32   [#permalink] 12 Sep 2017, 06:17
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# M26-32

Moderators: Bunuel, chetan2u

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