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