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

Find the value of the expression 2-(-2)⁻².

Pick an answer.

(A)

-2

(B)

$\dfrac{1}{16}$

(C)

$\dfrac{7}{4}$

(D)

$\dfrac{9}{4}$

(E)

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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 $\frac{1}{4}$ , not $\frac{1}{16}$ (kills (B)).

1STEP 1

Turn the negative exponent into a reciprocal

A negative exponent doesn't make a negative number — it flips the power to the bottom, so (-2)⁻² becomes $\frac{1}{(-2)²}$ .

(-2)⁻² =

\frac{1}{(-2)²}

2STEP 2

Square the base

Squaring means multiplying by itself, and a negative times a negative is positive: (-2)² = 4, so the power equals $\frac{1}{4}$ .

(-2)² = (-2)·(-2) = 4 → (-2)⁻² =

\frac{1}{4}

3STEP 3

Do the subtraction

Put $\frac{1}{4}$ back in and rename 2 as $\frac{8}{4}$ to match denominators: $\frac{8}{4}$ - $\frac{1}{4}$ = $\frac{7}{4}$ , choice (C).

2-(-2)⁻² = 2-

\frac{1}{4}

\frac{8}{4}

\frac{1}{4}

\frac{7}{4}

→ (C)

Answer

\frac{7}{4}

The subtracted term

\frac{1}{4}

is small and positive, so the answer should sit just below 2 — and

\frac{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,

\frac{1}{16}

, gives choice (B). Avoiding both leaves

\frac{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

Turn the negative exponent into a reciprocal

Square the base

Do the subtraction

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