It is currently 16 Aug 2017, 08:12

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

Events & Promotions in June
Open Detailed Calendar

S96-16

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

Hide Tags

Expert Post
Math Expert
User avatar
D
Joined: 02 Sep 2009
Posts: 40882

Kudos [?]: 118267 [0], given: 11995

S96-16 [#permalink]

Show Tags

New post 16 Sep 2014, 01:51
Expert's post
3
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

35% (01:20) correct 65% (01:03) wrong based on 23 sessions

HideShow timer Statistics

The operation \(x N\) for all positive integers n greater than 1 is defined in the following manner:

\(x N = x\) raised to the power of \(x @ (n-1)\).

If \(x 1 = x\), which of the following expressions has the greatest value?


A. \((3 @ 2) @ 2\)
B. \(3 @ (1 @ 3)\)
C. \((2 @ 3) @ 2\)
D. \(2 @ (2 @ 3)\)
E. \((2 @ 2) @ 3\)
[Reveal] Spoiler: OA

_________________

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

Kudos [?]: 118267 [0], given: 11995

Expert Post
Math Expert
User avatar
D
Joined: 02 Sep 2009
Posts: 40882

Kudos [?]: 118267 [0], given: 11995

Re S96-16 [#permalink]

Show Tags

New post 16 Sep 2014, 01:51
Official Solution:


The operation \(x [b]N
\) for all positive integers n greater than 1 is defined in the following manner:

\(x N = x\) raised to the power of \(x @ (n-1)\).

If \(x 1 = x\), which of the following expressions has the greatest value?
[/b]

A. \((3 @ 2) @ 2\)
B. \(3 @ (1 @ 3)\)
C. \((2 @ 3) @ 2\)
D. \(2 @ (2 @ 3)\)
E. \((2 @ 2) @ 3\)


First, we need to figure out what this strange operation means for a few small integers n. Let's build upward from 1:
\(x 1 = x\)

\(x @ 2 = x\) raised to the power of \(x 1\) (which is just \(x\)), so \(x @ 2 = x^x\)

\(x @ 3 = x\) raised to the power of \(x @ 2\), so \(x @ 3 = x^{x^x}\)
\(x @ 4 = x^{x^{x^x}}\)

So the number after the @ sign tells you how many \(x\)'s are in the exponential expression. Now we can translate the answer choices. As always, do the operation inside the parentheses first.

(A) \((3 @ 2) @ 2\)
\(3 @ 2 = 3^3 = 27\)
\(27 @ 2 = 27^{27} = (3^3)^{27} = 3^{81}\)

(B) \(3 @ (1 @ 3)\)
\(1 @ 3 = 1^{1^1} = 1^1 = 1\)
\(3@1 = 3\)

(C) \((2 @ 3) @ 2\)
\(2 @ 3 = 2^{2^2} = 2^4 = 16\)
\(16 @ 2 = 16^{16} = (2^4)^{16} = 2^{64}\)

Because both the base and the exponent of this answer choice are smaller, we can tell that choice A is still the winner at this point.

(D) \(2 @ (2 @ 3)\)
\(2 @ 3 = 2^{2^2} = 2^4 = 16\)
\(2 16 = 2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^2}}}}}}}}}}}}}}\)

There are sixteen 2's in this "tower of powers"! This number is incredibly large, far larger than \(3^{81}\). Let's start to collapse the layers to see why.
\(2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^2}}}}}}}}}}}}}}\)
\(= 2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^4}}}}}}}}}}}}}\)
\(= 2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{16}}}}}}}}}}}}}\)

\(2^{16} = 65536\) You aren't expected to know that, of course, but now imagine 2 raised to that power. This number has thousands of digits.

Now imagine 2 raised to THAT power.

Then 2 raised to THAT power.

And so on, over 10 more times!

This number is the winner by far among the first four answer choices.

(E) \((2 @ 2) @ 3\)
\(2 @ 2 = 2^2 = 4\)
\(4 @ 3 = 4^{4^4} = 4^{256} = 2^{512}\)

While enormous, this number is still far smaller than answer choice (D).

By the way, the operation represented by the @ sign in this problem is sometimes called "tetration." The reason is that just as multiplication is repeated addition, and exponentiation is repeated multiplication, so-called "tetration" is repeated exponentiation. ("Tetra-" means "four," and this operation is fourth in line: addition, multiplication, exponentiation, tetration.) Tetration is also called superexponentiation, ultraexponentiation, hyper-4, and power tower.


Answer: D
_________________

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

Kudos [?]: 118267 [0], given: 11995

Manager
Manager
avatar
Joined: 03 Jan 2014
Posts: 65

Kudos [?]: 13 [0], given: 93

Concentration: Strategy, Operations
GMAT 1: 720 Q46 V42
GPA: 3.86
WE: Information Technology (Consulting)
Re: S96-16 [#permalink]

Show Tags

New post 14 Mar 2016, 19:38
Awesome explanation and cool question. I read about Graham's number recently and that talks about tetration and even higher powers of these kinds of operations. The numbers get so large that all the atoms in the universe aren't even close to come near an approximation. It's mind blowing. Here is the article: http://waitbutwhy.com/2014/11/1000000-grahams-number.html

Kudos [?]: 13 [0], given: 93

Intern
Intern
avatar
Joined: 12 Nov 2016
Posts: 11

Kudos [?]: 2 [0], given: 2

Re: S96-16 [#permalink]

Show Tags

New post 18 Dec 2016, 14:27
This is a great question, however, I think it allows for all values of n>1. I do not think we can use the value n=1.

So with the way the question is worded, I do not think we can use x@1=x

The operation x@n for all positive integers n greater than 1.

If this is changed to n greater than 0, it would be okay. Bunuel , what are your thoughts?

Kudos [?]: 2 [0], given: 2

Intern
Intern
avatar
B
Status: Single
Affiliations: None
Joined: 25 Dec 2016
Posts: 10

Kudos [?]: 6 [0], given: 37

Sricharan: A
Re S96-16 [#permalink]

Show Tags

New post 11 Jan 2017, 11:47
I think this is a high-quality question and I agree with explanation.

Kudos [?]: 6 [0], given: 37

Manager
Manager
User avatar
S
Joined: 31 Jan 2017
Posts: 57

Kudos [?]: 26 [0], given: 17

Location: India
Concentration: Strategy, Leadership
Schools: ISB '19, IIMA , IIMB, IIMC , XLRI, IIM
GMAT 1: 680 Q49 V34
GPA: 4
WE: Project Management (Energy and Utilities)
Premium Member Reviews Badge
Re: S96-16 [#permalink]

Show Tags

New post 16 Feb 2017, 07:37
The question is great.

Note: The question tells us that the operation is defined in a manner for the case: n greater than 1. But for case: n=1, the operation results x.
_________________

__________________________________
Kindly press "+1 Kudos" if the post helped :)

Kudos [?]: 26 [0], given: 17

Re: S96-16   [#permalink] 16 Feb 2017, 07:37
Display posts from previous: Sort by

S96-16

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

Moderator: Bunuel



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

Powered by phpBB © phpBB Group and phpBB SEO

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®.