Competition · AMC preparation · step 4 of 4
AMC 10 · 2019A · #1
Grade 6 arithmeticPick an answer.
AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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^anything = 1 and 2⁰ = 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).
Split into two pieces
Split into two pieces: L = 2^{(0^(1⁹))} and R = ((2⁰)¹)⁹, solve each, then add.
Splitting a long expression into named pieces makes the order of operations obvious.
5.OA.A.1Identify SubproblemsEvaluate the left tower
Innermost first: 1⁹ = 1, then 0¹ = 0, then 2⁰ = 1, so L = 1.
Two repeated patterns: 1 to any power is 1, and 2⁰ = 1.
One raised to any power stays one, and any nonzero number to the zero power is also one.
▸ Why?
An exponent counts how many times a factor is used, and using one over and over changes nothing.
▸ Why?
One is the number that leaves everything unchanged when multiplied, which is why it is the empty product.
Evaluate the right tower
Same way: 2⁰ = 1, then 1¹ = 1, then 1⁹ = 1, so R = 1.
Same pattern as the left side — once you hit 1, any further exponent keeps it 1.
6.EE.A.1Look For A PatternAdd the two pieces
Add the pieces: L + R = 1 + 1 = 2, which is choice (C).
Combining the two subproblem answers gives the final value.
1.OA.C.6Identify SubproblemsCheck against the choices
Elimination check: two pieces each equal to 1 can only sum to 2, so (C).
Two ones can only sum to two — no other choice fits.
6.EE.A.1Eliminate PossibilitiesThis AMC 10 problem only needs Grade 6 "evaluate expressions with whole-number exponents" you already know — every 1 to a power stays 1, every 2⁰ is 1, so each piece is 1 and the sum is 2.
- Split into two pieces
- Evaluate the left tower
- Evaluate the right tower
- Add the two pieces
- Check against the choices
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