S96-16

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