Printable

2019 AMC 10A Problem 1

Problem only Back to the problem

2019 AMC 10A Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Exponents, Logarithms & Radicals · difficulty 1 of 5 · about 1 min · position P1-10

What is the value of $2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?$

A)
$0$
B)
$1$
C)
$2$
D)
$3$
E)
$4$

Answer

Key insight

Evaluate the towers from the top down: 1^9 = 1 makes 0^1 = 0, so the first term is 2^0 = 1; the second is a power of 1.

Solution

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 $\boxed{\textbf{(C)}\ 2}$ .

Why this works

A tower of exponents is evaluated from the top down, and once the inner pieces collapse to $0$ or $1$ the rest is two facts: a nonzero base to the power $0$ is $1$ , and $0$ to a positive power is $0$ . Slow down and write each layer; the problem is designed so that a rushed reader swaps those two facts.

The trap

Treating 0 raised to a power as 1 (confusing it with 2^0), which gives 3 or 4.

Common mistakes

Treating 0 raised to a power as 1 (confusing it with 2^0), which gives 3 or 4.
Reading $2^{0^{1^9}}$ as $(2^0)^{1^9}$ and losing track of which exponent applies to which base.

Techniques

Set up the equation/formula and compute; no special trick needed