AMC 10 · 2013 · #8
Grade 8 arithmeticWhat is the value of \frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?
Pick an answer.
AMC 10 2013 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 $\dfrac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$.
Givens: Numerator is $2^{2014}+2^{2012}$; Denominator is $2^{2014}-2^{2012}$; Answer choices: (A) $-1$, (B) $1$, (C) $\frac{5}{3}$, (D) $2013$, (E) $2^{4024}$
Unknowns: The single number the fraction equals
Understand
Restated: Find the value of the fraction $\dfrac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$.
Givens: Numerator is $2^{2014}+2^{2012}$; Denominator is $2^{2014}-2^{2012}$; Answer choices: (A) $-1$, (B) $1$, (C) $\frac{5}{3}$, (D) $2013$, (E) $2^{4024}$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #9 Solve an Easier Related Problem, #7 Identify Subproblems
The giant exponents make the fraction look scary, but every term is built from the same block $2^{2012}$. Tool #4 (Introduce a Variable) names that block $x = 2^{2012}$, which instantly shrinks the monster into the easy fraction $\frac{4x+x}{4x-x}$ (Tool #9, Solve an Easier Related Problem). Tool #7 then splits the work into two small subproblems: rewrite the top and bottom, then cancel the shared $x$.
Execute — Answer: C
8.EE.A.1 Step 1 Name the common block
- Let $x = 2^{2012}$ stand for the smaller power.
- Because $2014 = 2012 + 2$, the larger power is $2^{2014} = 2^{2012} \cdot 2^{2} = 4x$.
- This uses the exponent rule that adding exponents means multiplying powers of the same base.
💡 Two powers of the same base only differ by how many extra factors of the base you have, so $2^{2014}$ is just $2^{2012}$ with two more $2$'s stuck on.
6.EE.A.3 Step 2 Rewrite top and bottom
- Replace every power with its $x$-form.
- The numerator becomes $4x + x$ and the denominator becomes $4x - x$.
- Combining like terms turns them into $5x$ and $3x$.
💡 Once both powers are written with the same block $x$, they are just like terms you can add or subtract like $4$ apples plus $1$ apple.
7.NS.A.2 Step 3 Cancel the shared factor
- The factor $x = 2^{2012}$ appears in both the top and the bottom, and it is not zero, so it divides out.
- That leaves the plain fraction $\frac{5}{3}$, which matches choice (C).
💡 Multiplying top and bottom of a fraction by the same nonzero number never changes its value, so dividing that number back out is free.
8.EE.A.1 Let $x = 2^{2012}$ stand for the smaller power. Because $2014 = 2012 + 2$, the l 6.EE.A.3 Replace every power with its $x$-form. The numerator becomes $4x + x$ and the de 7.NS.A.2 The factor $x = 2^{2012}$ appears in both the top and the bottom, and it is not Review
Reasonableness: Sanity check the size: the numerator $2^{2014}+2^{2012}$ is a bit bigger than the denominator $2^{2014}-2^{2012}$, so the fraction must be slightly more than $1$. That rules out (A) $-1$, (B) $1$, the huge (D) $2013$, and the astronomical (E) $2^{4024}$, leaving only (C) $\frac{5}{3}\approx 1.67$. Exact recheck: factor $2^{2012}$ straight out to get $\frac{2^{2012}(2^2+1)}{2^{2012}(2^2-1)} = \frac{5}{3}$.
Alternative: Skip the variable and factor $2^{2012}$ directly from numerator and denominator: $\frac{2^{2012}(2^{2}+1)}{2^{2012}(2^{2}-1)} = \frac{4+1}{4-1} = \frac{5}{3}$. Same answer, no substitution needed.
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Rewriting $2^{2014}$ as $2^{2012}\cdot 2^{2} = 4x$ so both powers share the common block $x = 2^{2012}$.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Combining like terms $4x + x = 5x$ and $4x - x = 3x$ in the numerator and denominator.)7.NS.A.2Apply and extend understanding of multiplication and division of rational numbers (Dividing out the common nonzero factor $x$ from $\frac{5x}{3x}$ to reach $\frac{5}{3}$.)
⭐ When huge powers share a common block, name it $x$ — the scary fraction $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$ collapses to $\frac{5x}{3x} = \frac{5}{3}$.
⭐ When huge powers share a common block, name it $x$ — the scary fraction $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$ collapses to $\frac{5x}{3x} = \frac{5}{3}$.
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