GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 20 Nov 2018, 20:08

Close

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
Your Progress

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.

Close

Request Expert Reply

Confirm Cancel
Events & Promotions in November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
  • All GMAT Club Tests are Free and open on November 22nd in celebration of Thanksgiving Day!

     November 22, 2018

     November 22, 2018

     10:00 PM PST

     11:00 PM PST

    Mark your calendars - All GMAT Club Tests are free and open November 22nd to celebrate Thanksgiving Day! Access will be available from 0:01 AM to 11:59 PM, Pacific Time (USA)
  • Free lesson on number properties

     November 23, 2018

     November 23, 2018

     10:00 PM PST

     11:00 PM PST

    Practice the one most important Quant section - Integer properties, and rapidly improve your skills.

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Manager
Manager
User avatar
Joined: 13 Jul 2010
Posts: 139
For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

Show Tags

New post 04 Oct 2010, 18:14
3
14
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

54% (01:59) correct 46% (02:05) wrong based on 378 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
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50711
Re: functions  [#permalink]

Show Tags

New post 05 Oct 2010, 02:04
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: \(2^{2*3}=2^6=64\neq2^{2^3}=2^8=256\).

Answer: B (II only).

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.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

General Discussion
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 769
Location: London
GMAT ToolKit User Reviews Badge
Re: functions  [#permalink]

Show Tags

New post 04 Oct 2010, 22:39
Not sure if this question is copied correctly, almost looks like it should be "which is not always true"
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

GMAT Club Premium Membership - big benefits and savings

Manager
Manager
User avatar
Joined: 13 Jul 2010
Posts: 139
Re: functions  [#permalink]

Show Tags

New post 06 Oct 2010, 18:02
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!
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 769
Location: London
GMAT ToolKit User Reviews Badge
Re: functions  [#permalink]

Show Tags

New post 06 Oct 2010, 23:48
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
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

GMAT Club Premium Membership - big benefits and savings

Intern
Intern
avatar
Joined: 20 Nov 2013
Posts: 25
Schools: LBS '17
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

Show Tags

New post 10 Feb 2014, 22:43
The question is written wrong. Can this be written correctly please !
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50711
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

Show Tags

New post 10 Feb 2014, 22:48
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 8867
Premium Member
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab  [#permalink]

Show Tags

New post 25 Aug 2018, 06:58
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).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

GMAT Club Bot
Re: For all real numbers a and b, where a ⋅ b =/ 0, let a◊b = ab &nbs [#permalink] 25 Aug 2018, 06:58
Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.