AMC 10 · 2008 · #7
Grade 8 algebraThe fraction
(32007)2−(32005)2(32008)2−(32006)2
simplifies to which of the following?
Pick an answer.
AMC 10 2008 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: A fraction has a difference of two squares on top and another difference of two squares on the bottom, all built from powers of 3. Simplify it to a single value.
Givens: Numerator: $\left(3^{2008}\right)^2-\left(3^{2006}\right)^2$; Denominator: $\left(3^{2007}\right)^2-\left(3^{2005}\right)^2$; Every term is a power of 3, and the answer must be one of the five choices.
Unknowns: The single simplified value of the whole fraction.
Understand
Restated: A fraction has a difference of two squares on top and another difference of two squares on the bottom, all built from powers of 3. Simplify it to a single value.
Givens: Numerator: $\left(3^{2008}\right)^2-\left(3^{2006}\right)^2$; Denominator: $\left(3^{2007}\right)^2-\left(3^{2005}\right)^2$; Every term is a power of 3, and the answer must be one of the five choices.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #7 Identify Subproblems, #9 Solve an Easier Related Problem
The exponents (2008, 2007, ...) are only there to scare you. All four powers sit close together, so give the smallest one a name, $a = 3^{2005}$, and rewrite every other power as a small multiple of $a$. The giant expression collapses into a plain fraction of two-digit numbers that you can divide by hand.
Execute — Answer: E
6.EE.B.6 Step 1 Name the smallest power
- Let $a = 3^{2005}$, the smallest of the four powers.
- Everything in the fraction is a power of 3, so each term can be written using this one block instead of a huge exponent.
💡 One name for the scary part turns four monster powers into one manageable letter.
8.EE.A.1 Step 2 Rewrite each power
- Since powers of the same base add exponents, $3^{2006}=3^1\cdot a=3a$, $3^{2007}=3^2\cdot a=9a$, and $3^{2008}=3^3\cdot a=27a$.
- Now square each one: $\left(3^{2008}\right)^2=(27a)^2=729a^2$, $\left(3^{2006}\right)^2=(3a)^2=9a^2$, $\left(3^{2007}\right)^2=(9a)^2=81a^2$, and $\left(3^{2005}\right)^2=a^2$.
💡 Adding exponents lets a power be split into a small factor times the common block.
6.EE.A.3 Step 3 Combine top and bottom
- Handle the numerator and denominator separately.
- Top: $729a^2-9a^2=720a^2$.
- Bottom: $81a^2-a^2=80a^2$.
- Each is now a single term times $a^2$.
💡 Like terms with the same $a^2$ subtract just like ordinary numbers.
6.EE.A.3 Step 4 Cancel and divide
- The factor $a^2$ appears on top and bottom, so it cancels, leaving $\frac{720}{80}=9$.
- So the fraction simplifies to 9, which is answer (E).
💡 The same factor on top and bottom divides out, and only a small quotient is left.
6.EE.B.6 Let $a = 3^{2005}$, the smallest of the four powers. Everything in the fraction 8.EE.A.1 Since powers of the same base add exponents, $3^{2006}=3^1\cdot a=3a$, $3^{2007} 6.EE.A.3 Handle the numerator and denominator separately. Top: $729a^2-9a^2=720a^2$. Bott 6.EE.A.3 The factor $a^2$ appears on top and bottom, so it cancels, leaving $\frac{720}{8 Review
Reasonableness: The numerator's powers are each two exponents higher than the matching powers in the denominator, so the whole fraction should scale by $3^2=9$. The clean result 9 lands exactly on choice (E), and 9 is one of the listed options, so nothing was dropped.
Alternative: Skip the variable and factor common powers directly. Using $\left(3^{n}\right)^2=3^{2n}$, the numerator is $3^{4016}-3^{4012}=3^{4012}(3^4-1)$ and the denominator is $3^{4014}-3^{4010}=3^{4010}(3^4-1)$. The $(3^4-1)=80$ factor cancels, leaving $\frac{3^{4012}}{3^{4010}}=3^2=9$.
CCSS standards used (min grade 8)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the smallest power $a=3^{2005}$ so the giant exponents become a single letter.)8.EE.A.1Know and apply the properties of integer exponents (Splitting each power into a small factor times $a$ and squaring it with the power rules.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Subtracting like $a^2$ terms and canceling the common $a^2$ factor to reach 720/80.)
⭐ When exponents look gigantic, give the smallest power a nickname; the monster numbers shrink into a fraction you can divide in your head.
⭐ When exponents look gigantic, give the smallest power a nickname; the monster numbers shrink into a fraction you can divide in your head.
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