AMC 10 · 2015 · #1
Grade 8 arithmeticPick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The expression stacks several small operations — two powers, a chain of add/subtract inside parentheses, a -1 exponent, and a final multiplication — so Tool #7 (Identify Subproblems) handles them one layer at a time, inside-out. Tool #3 (Eliminate Possibilities) guards against the built-in traps: the -1 exponent does not make the answer negative (kills the large negatives (A) and (B)), and forgetting the reciprocal-and-multiply leaves a bare 25 (kills (E)).
Evaluate the two powers
Evaluate each power: 2⁰=1 (not 0) and 5²=25.
A zero exponent always gives 1; squaring just means multiply the number by itself.
8.EE.A.1Identify SubproblemsSimplify inside the parentheses
Left to right, the parentheses simplify to 25.
Finish everything inside the parentheses before touching the exponent outside them.
5.OA.A.1Identify SubproblemsApply the -1 exponent
The ⁻¹ exponent means reciprocal, so 25⁻¹=1/25 — not a sign flip.
Raising to the power -1 just turns a number into one over itself.
8.EE.A.1Identify SubproblemsMultiply by 5 and simplify
Multiplying by 5 and cancelling gives 5/25=1/5 — choice (C).
Multiplying a fraction by a whole number scales the numerator, then cancel the shared factor.
5.NF.B.4Eliminate PossibilitiesSimplify everything inside the parentheses first; a -1 exponent just flips the number into one over itself, it never makes it negative.
- Evaluate the two powers
- Simplify inside the parentheses
- Apply the -1 exponent
- Multiply by 5 and simplify