GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 28 Mar 2020, 06:15 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab

Author Message
TAGS:

### Hide Tags

Manager  Joined: 13 Jul 2010
Posts: 97
For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

3
19 00:00

Difficulty:   85% (hard)

Question Stats: 51% (01:59) correct 49% (02:02) wrong based on 360 sessions

### HideShow timer Statistics

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

Source: Nova GMAT
Difficulty Level: 700

Originally posted by gettinit on 04 Oct 2010, 18:14.
Last edited by SajjadAhmad on 04 Jul 2019, 00:08, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 62285
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

4
5
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.
_________________
##### General Discussion
Retired Moderator Joined: 02 Sep 2010
Posts: 706
Location: London
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

1
Not sure if this question is copied correctly, almost looks like it should be "which is not always true"
_________________
Manager  Joined: 13 Jul 2010
Posts: 97
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

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!
Retired Moderator Joined: 02 Sep 2010
Posts: 706
Location: London
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

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
_________________
Intern  Joined: 08 Aug 2018
Posts: 1
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

### Show Tags

Hi,

I still don't understand statement 2. Why is it always true Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab   [#permalink] 26 Feb 2020, 06:53
Display posts from previous: Sort by

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