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# For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab

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Manager
Joined: 13 Jul 2010
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For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

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04 Oct 2010, 19:14
3
14
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Difficulty:

75% (hard)

Question Stats:

54% (01:59) correct 46% (02:04) wrong based on 369 sessions

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For all real numbers $$a$$ and $$b$$, where $$ab\neq{0}$$, let $$a@b=a^b$$. Then which of the following MUST be true?

I. $$a@b=b@a$$

II. $$(-a)@(-a) =\frac{(-1)^{-a}}{a^a}$$

III. $$(a@b)@c=a@(b@c)$$

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
Math Expert
Joined: 02 Sep 2009
Posts: 49430

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05 Oct 2010, 03:04
3
4
gettinit wrote:
For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab . Then which of the
following must be true?
I. a◊b = b◊a
II. (−a)◊(−a)= (−1)^−a / a^a
III. ( a◊b)◊c = a◊(b◊c)
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only

Please explain the math in II in detail. Thank you.

The question is as follows:

For all real numbers $$a$$ and $$b$$, where $$ab\neq{0}$$, let $$a@b=a^b$$. Then which of the following MUST be true?

I. $$a@b=b@a$$

II. $$(-a)@(-a) =\frac{(-1)^{-a}}{a^a}$$

III. $$(a@b)@c=a@(b@c)$$

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only

I. $$a@b=b@a$$ --> $$a@b=a^b$$ and $$b@a=b^a$$, these 2 expressions are not always equal: $$2^3\neq{3^2}$$;

II. $$(-a)@(-a)=\frac{(-1)^{-a}}{a^a}$$ --> $$(-a)@(-a)=(-a)^{-a}=(-1*a)^{-a}=-1^{-a}*a^{-a}=\frac{-1^{-a}}{a^a}$$ --> $$\frac{-1^{-a}}{a^a}=\frac{-1^{-a}}{a^a}$$, so this statement is always true;

III. $$(a@b)@c=a@(b@c)$$ --> $$(a@b)@c=(a^b)^c=a^{bc}$$ and $$a@(b@c)=a^{(b^c)}=a^{b^c}$$ these 2 expressions are not always equal: $$2^{2*3}=2^6=64\neq2^{2^3}=2^8=256$$.

Notes for III:

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
$$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$, which on the other hand equals to $$a^{mn}$$.

So:
$$(a^m)^n=a^{mn}$$;

$$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$.

Hope it helps.
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##### General Discussion
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04 Oct 2010, 23:39
Not sure if this question is copied correctly, almost looks like it should be "which is not always true"
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Joined: 13 Jul 2010
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06 Oct 2010, 19:02
Thanks Bunuel very informative and helpful. One question as I am new here, how did you get the symbols to show up in the question? I obviously did not know how to do this.

Kudos for you my friend!
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Joined: 02 Sep 2010
Posts: 772
Location: London

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07 Oct 2010, 00:48
gettinit wrote:
Thanks Bunuel very informative and helpful. One question as I am new here, how did you get the symbols to show up in the question? I obviously did not know how to do this.

Kudos for you my friend!

http://gmatclub.com/forum/writing-mathematical-symbols-in-posts-72468.html#p536379
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Joined: 20 Nov 2013
Posts: 26
Schools: LBS '17
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

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10 Feb 2014, 23:43
The question is written wrong. Can this be written correctly please !
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Joined: 02 Sep 2009
Posts: 49430
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

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10 Feb 2014, 23:48
amz14 wrote:
The question is written wrong. Can this be written correctly please !

Done. Edited the original post.
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Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

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25 Aug 2018, 07:58
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab &nbs [#permalink] 25 Aug 2018, 07:58
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