AMC 10 · 2015 · #1

Grade 8 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
exponentsorder-of-operationsfraction-arithmetic identify-subproblems ↑ Prerequisites: exponents
📏 Short solution 💡 1 insight
Problem
A negative base carries a negative exponent inside a subtraction. Find the value.

Pick an answer.

(A)
-2
(B)
$\dfrac{1}{16}$
(C)
$\dfrac{7}{4}$
(D)
$\dfrac{9}{4}$
(E)
6
How to solve
Strategy Identify Subproblems

The expression mixes two tricky pieces — a negative exponent and a negative base — so Tool #7 (Identify Subproblems) handles them one at a time: first turn the negative exponent into a reciprocal, then square the base, then do the single subtraction. Tool #3 (Eliminate Possibilities) guards against the classic traps: a negative exponent does not make the answer negative (kills (A)), and an even power of -2 is positive, so the term is exactly 1/4, not 1/16 (kills (B)).

1STEP 1

Turn the negative exponent into a reciprocal

The negative exponent means a reciprocal.

(-2)⁻² = 1/((-2)²)
2STEP 2

Square the base

Squaring the base makes it positive.

(-2)² = (-2)·(-2) = 4 → (-2)⁻² = 1/4
3STEP 3

Do the subtraction

Subtracting gives 7/4, choice (C).

2-(-2)⁻² = 2-1/4 = 8/4-1/4 = 7/4 → (C)
Answer
7/4
The subtracted term 1/4 is small and positive, so the answer should sit just below 2 — and 7/4 = 1.75 does. The two common mistakes both land on wrong choices: reading the negative exponent as a negative number gives something near -2 (choice (A)), and squaring only the 2 while forgetting it is already in a reciprocal, 1/16, gives choice (B). Avoiding both leaves 7/4.
💡Key takeaway

A negative exponent flips the power into a fraction's bottom — it does not make the number negative — so handle the exponent first, then subtract.

  • Turn the negative exponent into a reciprocal
  • Square the base
  • Do the subtraction

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