AMC 10 · 2008 · #7
Grade 8 algebraPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The exponents (2008, 2007, ...) are only there to scare you. All four powers sit close together, so give the smallest one a name, a = 3²⁰⁰⁵, 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.
Name the smallest power
Let a = 3²⁰⁰⁵, the smallest of the four powers, so every term becomes a small multiple of one block.
One name for the scary part turns four monster powers into one manageable letter.
6.EE.B.6Introduce A VariableRewrite each power
Same base adds exponents, so 3²⁰⁰⁶=3a, 3²⁰⁰⁷=9a, 3²⁰⁰⁸=27a; squaring gives 729a², 9a², 81a², a².
Adding exponents lets a power be split into a small factor times the common block.
Adding exponents lets each power be split into a small factor times one common block.
▸ Why?
An exponent counts how many times a factor is used, so a bigger exponent is the block used a few more times.
▸ Why?
That shared block can then be lifted out of the whole sum at once.
Combine top and bottom
Subtract the like terms separately: the top becomes 720a² and the bottom becomes 80a².
Like terms with the same a² subtract just like ordinary numbers.
6.EE.A.3Identify SubproblemsCancel and divide
The a² cancels top and bottom, leaving 720 divided by 80, which is 9 — choice (E).
The same factor on top and bottom divides out, and only a small quotient is left.
6.EE.A.3Introduce A VariableWhen exponents look gigantic, give the smallest power a nickname; the monster numbers shrink into a fraction you can divide in your head.
- Name the smallest power
- Rewrite each power
- Combine top and bottom
- Cancel and divide