AMC 10 · 2019 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

exponentsorder-of-operationsidentify-subproblems identify-subproblemspattern-recognition ↑ Prerequisites: exponentsorder-of-operations

📏 Short solution 💡 2 insights

Problem

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

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Compute $2^{\left(0^{\left(1^9\right)}\right)} + \left(\left(2^0\right)^1\right)^9$ — a stacked tower of exponents on the left and a chain of exponents on the right.

Givens: Left term: $2$ raised to $0^{(1^9)}$; Right term: $((2^0)^1)^9$; Answer choices: (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

Unknowns: The value of the whole expression

Understand

Restated: Compute $2^{\left(0^{\left(1^9\right)}\right)} + \left(\left(2^0\right)^1\right)^9$ — a stacked tower of exponents on the left and a chain of exponents on the right.

Givens: Left term: $2$ raised to $0^{(1^9)}$; Right term: $((2^0)^1)^9$; Answer choices: (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities

Tool #7 (Subproblems): the expression is a sum of two pieces — solve each piece separately and add. Inside each piece, evaluate exponents innermost-first, one tiny step at a time. Tool #5 (Pattern): the recurring facts $1^{\text{anything}} = 1$ and $2^0 = 1$ do almost all the work. Tool #3 (Eliminate): once we notice each piece is $1$, the sum can only be $2$, ruling out (A), (B), (D), (E).

Execute — Answer: C

#7 Identify Subproblems 5.OA.A.1 Step 1

Break the expression into two subproblems: $L = 2^{\left(0^{\left(1^9\right)}\right)}$ and $R = \left(\left(2^0\right)^1\right)^9$.
We will compute each, then add.

$L + R$ where $L = 2^{\left(0^{\left(1^9\right)}\right)},\ R = \left(\left(2^0\right)^1\right)^9$

💡 Splitting a long expression into named pieces makes the order of operations obvious.

#5 Look for a Pattern 6.EE.A.1 Step 2

Evaluate $L$ from the innermost exponent outward.
First $1^9 = 1$ because $1$ raised to any power is $1$.
Then $0^1 = 0$.
Finally $2^0 = 1$.

$1^9 = 1 \;\Rightarrow\; 0^{1} = 0 \;\Rightarrow\; 2^{0} = 1$, so $L = 1$

💡 Two repeated patterns: $1$ to any power is $1$, and $2^0 = 1$.

#5 Look for a Pattern 6.EE.A.1 Step 3

Evaluate $R$ the same way, innermost first.
$2^0 = 1$.
Then $1^1 = 1$.
Then $1^9 = 1$.

$2^{0} = 1 \;\Rightarrow\; 1^{1} = 1 \;\Rightarrow\; 1^{9} = 1$, so $R = 1$

💡 Same pattern as the left side — once you hit $1$, any further exponent keeps it $1$.

#7 Identify Subproblems 1.OA.C.6 Step 4

Add the two pieces: $L + R = 1 + 1 = 2$.
This matches choice (C).

$$L + R = 1 + 1 = 2 \;\Rightarrow\; \textbf{(C)}$$

💡 Combining the two subproblem answers gives the final value.

#3 Eliminate Possibilities 6.EE.A.1 Step 5

Sanity check with elimination.
Both pieces are at most a few, and we computed each to be exactly $1$.
The only choice that equals $1 + 1$ is (C) $2$.

$$L = 1,\ R = 1 \;\Rightarrow\; 2 = \textbf{(C)}$$

💡 Two ones can only sum to two — no other choice fits.

[1] #7 5.OA.A.1 Break the expression into two subproblems: $L = 2^{\left(0^{\left(1^9\right)}\ri

[2] #5 6.EE.A.1 Evaluate $L$ from the innermost exponent outward. First $1^9 = 1$ because $1$ ra

[3] #5 6.EE.A.1 Evaluate $R$ the same way, innermost first. $2^0 = 1$. Then $1^1 = 1$. Then $1^9

[4] #7 1.OA.C.6 Add the two pieces: $L + R = 1 + 1 = 2$. This matches choice (C).

[5] #3 6.EE.A.1 Sanity check with elimination. Both pieces are at most a few, and we computed ea

Review

Reasonableness: Both pieces of the expression simplified independently to $1$, and $1 + 1 = 2$. The choices (A) $0$, (B) $1$, (D) $3$, (E) $4$ are impossible because two non-zero parts each equal to $1$ must sum to exactly $2$. (C) $2$ is consistent.

Alternative: Tool #5 (Look for a Pattern) alone is enough: every $1^k$ collapses to $1$ and every $2^0$ collapses to $1$, so each piece is just $1$. This is the fastest path — no formal algebra needed.

CCSS standards used (min grade 6)

1.OA.C.6 Add and subtract within 20 using strategies (Adding the two simplified pieces $1 + 1 = 2$.)
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Respecting the nested parentheses to evaluate the tower from the innermost exponent outward.)
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Computing each exponent step ($1^9 = 1$, $0^1 = 0$, $2^0 = 1$, $1^1 = 1$, $1^9 = 1$).)

⭐ This AMC 10 problem only needs Grade 6 "evaluate expressions with whole-number exponents" you already know — every $1$ to a power stays $1$, every $2^0$ is $1$, so each piece is $1$ and the sum is $2$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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