2019 AMC 10A Problem 1

A tower of exponents is worked out from the top down, so start with the smallest, innermost exponent in each term.

First term: $2^{\left(0^{\left(1^9\right)}\right)}$.

1. Innermost: $1^9 = 1$ (one times itself any number of times is still one).
2. Next layer: $0^1 = 0$.
3. Last layer: $2^0 = 1$ (any nonzero number to the power $0$ is $1$).

So the first term is $1$.

Second term: $\left(\left(2^0\right)^1\right)^9$.

1. Innermost: $2^0 = 1$.
2. Then $1^1 = 1$, and then $1^9 = 1$.

So the second term is also $1$.

Add: $1 + 1 = 2$.

The answer is $\boxed{\textbf{(C)}\ 2}$.

The whole problem rests on two small rules that look alike: $0$ raised to a positive power is $0$, while a nonzero number raised to the power $0$ is $1$. The problem is built so that a rushed reader swaps them. Writing one layer per line, from the top of the tower down, keeps them straight.

Work each term from the innermost exponent outward.

First term: $1^9 = 1$, so the exponent of $0$ is $1$ and $0^1 = 0$. That leaves $2^0 = 1$.

Second term: $2^0 = 1$, then $1^1 = 1$, then $1^9 = 1$.

Adding, $1 + 1 = 2$.

The answer is $\boxed{\textbf{(C)}\ 2}$.