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# Each of the following equations has at least one solution

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Re: No solution N: Manhattan GMAT test [#permalink]  05 May 2011, 22:42
for B and C n = 0
for D and E n = 1.

A prevails.
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Re: Each of the following has at least one solution EXCEPT [#permalink]  18 Aug 2012, 15:14
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
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Each of the following has at least one solution EXCEPT [#permalink]  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
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Re: Each of the following equations has at least one solution [#permalink]  08 Oct 2012, 04:00
if -2^n means -(2^n), the answer is A. Otherwise I find 0 to satisfy all equations.
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Re: Exponents [#permalink]  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 $$(-2)^{-n}=(-2)^{0}=1$$.

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?

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Re: Exponents [#permalink]  14 Oct 2012, 22:12
Expert's post
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 $$(-2)^{1/n}$$
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 03 Dec 2012 Posts: 367 Followers: 0 Kudos [?]: 67 [0], given: 291 Re: Each of the following equations has at least one solution [#permalink] 26 Oct 2013, 03:40 Can somebody please explain if (–2)^n = 1 or -1 (if n=0) SVP Joined: 06 Sep 2013 Posts: 2045 Concentration: Finance GMAT 1: 770 Q0 V Followers: 34 Kudos [?]: 370 [0], given: 355 Re: Exponents [#permalink] 15 Jan 2014, 08:57 nverma wrote: 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 Why did you plug n=1 for the last two, wouldn't it be easier just to plug n=0 for all and see that A has no solution? Just want to know if there was any specific reason why you did so Thank you Cheers J PS. Would be nice if we could get this question in code format! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5967 Location: Pune, India Followers: 1527 Kudos [?]: 8428 [0], given: 193 Re: Exponents [#permalink] 15 Jan 2014, 19:55 Expert's post jlgdr wrote: nverma wrote: 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 Why did you plug n=1 for the last two, wouldn't it be easier just to plug n=0 for all and see that A has no solution? Just want to know if there was any specific reason why you did so Thank you Cheers J PS. Would be nice if we could get this question in code format! We need to find the equation that has no solution. What we are trying to do is find at least one solution for 4 equations. The fifth one will obviously not have any solution and will be our answer. Options (D) and (E) do not have 0 as a solution. So you try n = 1 on (A), (D) and (E). n = 1 is still not a solution for (A) but it is for (D) and (E). (D) (–2)^n = –2^n When you put n = 0, you get (-2)^0 = -2^0 1 = -1 which doesn't hold. So you try n = 1 (–2)^1 = -2^1 -2 = -2 n = 1 is a solution. Same logic for (E) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: No solution N: Manhattan GMAT test [#permalink]  24 Aug 2014, 15:52
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Re: Each of the following equations has at least one solution [#permalink]  08 Sep 2015, 07:44
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Re: Each of the following equations has at least one solution   [#permalink] 08 Sep 2015, 07:44

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