AMC 10 · 2015 · #1
Grade 8 arithmeticPick an answer.
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
The zero power is 1, not zero.
A zero exponent always gives 1; squaring just means multiply the number by itself.
8.EE.A.1Identify SubproblemsSimplify inside the parentheses
Inside the brackets everything reduces to 25.
Finish everything inside the parentheses before touching the exponent outside them.
5.OA.A.1Identify SubproblemsApply the -1 exponent
The exponent flips that number.
Raising to the power -1 just turns a number into one over itself.
Raising to the power minus one just turns a number into one over itself.
▸ Why?
A number multiplied by its reciprocal gives one, so the reciprocal is exactly what undoes it.
▸ Why?
Landing on one is what marks the pair as undone, since multiplying by one changes nothing.
Multiply by 5 and simplify
Multiplying gives 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