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# Each of the following equations has at least one solution EXCEPT

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Manager
Joined: 28 Oct 2009
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Each of the following equations has at least one solution EXCEPT [#permalink]

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12 May 2010, 09:12
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Each of the following equations has at least one solution EXCEPT

A. $$–2^n = (–2)^{-n}$$

B. $$2^{-n} = (–2)^n$$

C. $$2^n = (–2)^{-n}$$

D. $$(–2)^n = –2^n$$

E. $$(–2)^{-n} = –2^{-n}$$
[Reveal] Spoiler: OA
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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12 May 2010, 10:53
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marcusaurelius wrote:
Each of the following equations has at least one solution EXCEPT

–2^n = (–2)^-n
2^-n = (–2)^n
2^n = (–2)^-n
(–2)^n = –2^n
(–2)^-n = –2^-n

IMHO A

a) –2^n = (–2)^-n
–2^n = 1/(–2)^n
–2^n * (–2)^n = 1, Keep it. Let's solve the other options..!!

b) 2^-n = (–2)^n
1/2^n = (–2)^n
1 = (–2)^n * (2^n)
For n=0, L.H.S = R.H.S

c) 2^n = (–2)^-n
2^n = 1/ (–2)^n
(2^n) * (–2)^n = 1
For n=0, L.H.S = R.H.S

d) (–2)^n = –2^n
(–2)^n / –2^n = 1
For n=1, L.H.S = R.H.S

e) (–2)^-n = –2^-n
1/ (–2)^n = 1/–2^n
For n=1, L.H.S = R.H.S
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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12 May 2010, 11:09
all seems to have n=0 as solution....? whats the OA
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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12 May 2010, 14:01
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gurpreetsingh wrote:
all seems to have n=0 as solution....? whats the OA

$$n=0$$ is not a solution of the equation $$-2^n = (-2)^{-n}$$ (in fact this equation has no solution):

$$-2^n=-(2^n)=-(2^{0})=-1$$ but $$(-2)^{-n}=(-2)^{0}=1$$.
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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12 May 2010, 15:13
yes right i didnt read it closely
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Intern
Joined: 15 Mar 2010
Posts: 25
Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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14 May 2010, 06:56
Hi,

it's really elegant question

0 is a solution for second and third equations.
1 is a solution for the last two equations.

So answer is A. Really if n is not equal to 0 then absolute value of left part is greater than 1 and right part is less than 1. In case when n is equal to 0 we will get -1=1.
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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14 May 2010, 12:49
great explanation Bunuel, thanks
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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15 Jun 2010, 13:15
Thanks for merging.

Do you mean a negative number raised to the power of 0 yields -1?? I didn't know that!
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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15 Jun 2010, 13:32
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study wrote:
Thanks for merging.

Do you mean a negative number raised to the power of 0 yields -1?? I didn't know that!

No.

Any number to the power of zero equals to 1 (except 0^0: 0^0 is undefined for GMAT and not tested).

The point here is that $$-2^n$$ means $$-(2^n)$$ and not $$(-2)^n$$. So for $$n=0$$ --> $$-2^n=-(2^n)=-(2^0)=-(1)$$. But if it were $$(-2)^n$$, then for $$n=0$$ --> $$(-2)^0=1$$.

Hope it's clear.
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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15 Jun 2010, 22:56
So how do you know that the point here is that -2^n means -(2^n) and not (-2)^n

The actual question has no parenthesis. This is tricky!
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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16 Jun 2010, 05:36
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study wrote:
So how do you know that the point here is that -2^n means -(2^n) and not (-2)^n

The actual question has no parenthesis. This is tricky!

I mean that $$-x^y$$ always means $$-(x^y)$$. If it's supposed to mean $$(-x)^y$$, then it would be represented this way.
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Re: Each of the following equations has at least one solution EXCEPT [#permalink]

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06 Dec 2010, 08:21
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SubratGmat2011 wrote:
Each of the following equations has at least one solution EXCEPT
-2^n = (-2)^-n
2^-n = (-2)^n
2^n = (-2)^-n
(-2)^n = -2^n
(-2)^-n = -2^-n

Can somebody plz help me out what is the approch for this type of problems?

The first and most straight forward approach that comes to mind is that I can see most of these equations will have n = 0 or n = 1 as a solution.
Except for the very first one:
n = 0: -2^0 = -1 while (-2)^(-0) = 1
n = 1: -2^1 = -2 while (-2)^-n = -1/2

For all other options, n = 0 or 1 satisfies the equation.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 17 Jan 2011 Posts: 227 Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 19 Jan 2011, 22:03 1 This post received KUDOS 1 This post was BOOKMARKED Lets look at each choice - A –2^n = (–2)^-n =>(-1).(2)^n = 1/(-2)^n =>(-1).(2)^n.(-2)^n = 1 =>(-1).(2)^n.(-1)^n.(2)^n = 1 =>(-1).(-1)^n.(2)^2n = 1 Above cannot be true for any value of n (No solution - answer) B 2^-n = (–2)^n =>1/(2)^n = (-2)^n =>1=(-1)^n.(2)^n.(2)^n =>1=(-1)^n.(2)^2n Above is true for n=0, so it has atleast one solution C 2^n = (–2)^-n =>(2)^n = 1/(-2)^n Rest of the steps Similar to option B D (–2)^n = –2^n =>(-1)^n. (2)^n = (-1).(2)^n =>(-1)^n = (-1) Above is true for all odd values of n E (–2)^-n = –2^-n =>1/[(-1)^n. (2)^n] = (-1)/(2)^n =>1/[(-1)^n] = (-1) =>1/(-1)^n = -1 Above is true for all odd values of n I hope this helps. _________________ Good Luck!!! ***Help and be helped!!!**** Senior Manager Joined: 08 Nov 2010 Posts: 385 WE 1: Business Development Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 18 Feb 2011, 10:22 i guess the only real way to solve it under 2 min is to plug in 0/1... if u start with choosing 1 here as first step is not good... choosing 0 is canceling 3 choices quickly... _________________ Director Joined: 01 Feb 2011 Posts: 703 Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 05 May 2011, 17:26 lets pick a value for n. n = 0 A. cannot be true as we get -1 on LHS and 1 on RHS ( as anything to the power of 0 is 1) B. true (LHS = RHS = 1) C. true (LHS = RHS = 1) D. true (LHS = RHS = 1) E. true (LHS = RHS = 1) Answer is A. VP Status: There is always something new !! Affiliations: PMI,QAI Global,eXampleCG Joined: 08 May 2009 Posts: 1260 Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 05 May 2011, 22:42 for B and C n = 0 for D and E n = 1. A prevails. _________________ Visit -- http://www.sustainable-sphere.com/ Promote Green Business,Sustainable Living and Green Earth !! Retired Moderator Affiliations: PMI, ASQ Joined: 16 Jun 2012 Posts: 119 GMAT 1: 710 Q49 V38 Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 18 Aug 2012, 15:14 1 This post was BOOKMARKED arichardson26 wrote: Each of the following has at least one solution EXCEPT A. -2^n = (-2)^-n B. 2^-n = (-2)^n C. 2^n = (-2)^-n D. (-2)^n = -2^n E. (-2)^-n = -2^-n [Reveal] Spoiler: A B, C have can be equated by using n=0 D and E have external/independent -ve signs, so 0 wont help, but using n= +1 for D and -1 for E will equate the sides. Took more than 2 mins _________________ Legendary Collections 1. 700 Level Quant 2. IIM Quant 3. 100 CR from LSAT 4. 100 Legendary SC 5. 5000 Practice problems 6.125 Quant 7. 38 SC 8. 10 Full Length GMAT Pen&Paper Tests 9. 1500+ RC 10. 100 Legendary CR 11. Additional Verbal Qs 12. Additional Quant Qs My debrief |Free essay review initiative PM me for One-on-One Webex Tutoring Senior Manager Joined: 24 Aug 2009 Posts: 489 Schools: Harvard, Columbia, Stern, Booth, LSB, Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 23 Aug 2012, 03:45 OA has to be A because Equation 1 simplifies to (2)^n (2)^n (-1)^n= -1 has no solution for any value of n Rest of options have at least 1 solution _________________ If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS. Kudos always maximizes GMATCLUB worth -Game Theory If you have any question regarding my post, kindly pm me or else I won't be able to reply Intern Joined: 16 Sep 2012 Posts: 15 Concentration: Finance, General Management GMAT 1: 710 Q47 V41 Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 14 Oct 2012, 07:05 Bunuel wrote: gurpreetsingh wrote: all seems to have n=0 as solution....? whats the OA $$n=0$$ is not a solution of the equation $$-2^n = (-2)^{-n}$$ (in fact this equation has no solution): $$-2^n=-(2^n)=-(2^{0})=-1$$ but $$(-2)^{-n}=(-2)^{0}=1$$. Thank you for your response. I would like to double check why we say that n=0 could be a solution in case of $$(-2)^{-n}$$ as $$(-2)^{-n} = (-2)^{1/n}$$ and then we can not divide by zero? Nik Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7942 Location: Pune, India Re: Each of the following equations has at least one solution EXCEPT [#permalink] ### Show Tags 14 Oct 2012, 22:12 NikRu wrote: I would like to double check why we say that n=0 could be a solution in case of $$(-2)^{-n}$$ as $$(-2)^{-n} = (-2)^{1/n}$$ and then we can not divide by zero? Nik $$(-2)^{-n} = 1/(-2)^n$$ not $$(-2)^{1/n}$$ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Each of the following equations has at least one solution EXCEPT   [#permalink] 14 Oct 2012, 22:12

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