AMC 10 · 2014 · #2
Grade 8 arithmeticWhat is 2−3+2−323+23?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Find the value of the fraction $\dfrac{2^3 + 2^3}{2^{-3} + 2^{-3}}$.
Givens: The top of the fraction is $2^3 + 2^3$; The bottom of the fraction is $2^{-3} + 2^{-3}$; A negative exponent means a reciprocal: $2^{-3} = \dfrac{1}{2^{3}}$; Answer choices: (A) $16$, (B) $24$, (C) $32$, (D) $48$, (E) $64$
Unknowns: The single number that the whole fraction equals
Understand
Restated: Find the value of the fraction $\dfrac{2^3 + 2^3}{2^{-3} + 2^{-3}}$.
Givens: The top of the fraction is $2^3 + 2^3$; The bottom of the fraction is $2^{-3} + 2^{-3}$; A negative exponent means a reciprocal: $2^{-3} = \dfrac{1}{2^{3}}$; Answer choices: (A) $16$, (B) $24$, (C) $32$, (D) $48$, (E) $64$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities
The fraction is really three small jobs stacked together, so Tool #7 (Identify Subproblems) tackles them one at a time: first add the two equal powers on top, then add the two equal reciprocals on the bottom, then divide. Tool #3 (Eliminate Possibilities) is a quick guard — the bottom is a tiny number ($\tfrac14$), so dividing by it must make the answer much bigger than the top value $16$, which rules out (A) $16$ immediately.
Execute — Answer: E
6.EE.A.1 Step 1 Add the two equal powers on top
- The top is $2^3 + 2^3$.
- Since $2^3 = 8$, this is $8 + 8 = 16$.
- Adding two copies of the same thing is doubling it, so the top equals $2 \cdot 2^3 = 2^4 = 16$.
💡 Two copies of a power add up to double that power, the same as bumping the exponent up by one.
8.EE.A.1 Step 2 Add the two equal reciprocals on the bottom
- The bottom is $2^{-3} + 2^{-3}$.
- A negative exponent means a reciprocal, so $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$.
- Adding the two copies gives $\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{2}{8} = \dfrac{1}{4}$.
💡 A negative exponent flips the power to the bottom of a fraction, turning $2^{-3}$ into $\tfrac18$.
6.NS.A.1 Step 3 Divide the top by the bottom
- Now the fraction is $\dfrac{16}{\,1/4\,}$.
- Dividing by $\dfrac14$ is the same as multiplying by its reciprocal $4$, so $16 \div \dfrac14 = 16 \cdot 4 = 64$.
- That matches choice (E), and the small denominator already ruled out (A).
💡 Dividing by a quarter counts how many quarters fit inside — four times as many as the number itself.
6.EE.A.1 The top is $2^3 + 2^3$. Since $2^3 = 8$, this is $8 + 8 = 16$. Adding two copies 8.EE.A.1 The bottom is $2^{-3} + 2^{-3}$. A negative exponent means a reciprocal, so $2^{ 6.NS.A.1 Now the fraction is $\dfrac{16}{\,1/4\,}$. Dividing by $\dfrac14$ is the same as Review
Reasonableness: The top is $16$ and the bottom is a small $\tfrac14$, so dividing must push the result well above $16$ — landing at $64$, four times the top, feels right. A student who forgot the negative exponent and read the bottom as $2^3 + 2^3 = 16$ too would get $16/16 = 1$, which is not even a choice, a clear signal that the reciprocal step matters.
Alternative: Keep everything in powers of $2$. The top is $2 \cdot 2^3 = 2^4$ and the bottom is $2 \cdot 2^{-3} = 2^{-2}$, so the fraction is $\dfrac{2^4}{2^{-2}} = 2^{4-(-2)} = 2^{6} = 64$, confirming (E).
CCSS standards used (min grade 8)
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating $2^3 = 8$ and adding $2^3 + 2^3 = 16$ on top of the fraction.)8.EE.A.1Know and apply the properties of integer exponents (Rewriting the negative exponent $2^{-3}$ as the reciprocal $\dfrac{1}{8}$ to add the bottom.)6.NS.A.1Interpret and compute quotients of fractions (Computing $16 \div \dfrac14 = 16 \cdot 4 = 64$ by multiplying by the reciprocal.)
⭐ Break a messy fraction into top, bottom, and divide — and remember a negative exponent flips the power into a fraction's bottom.
⭐ Break a messy fraction into top, bottom, and divide — and remember a negative exponent flips the power into a fraction's bottom.
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