Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
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 = xx = x^x (we’ll use the caret symbol ^ to represent exponentiation, since as we’ll see, we’re going to do it a lot!)
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 = 65,536. 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.
The correct answer is (D).