AMC 10 · 2009 · #6
Grade 8 algebraPick an answer.
Everything is written with the two building blocks P = 2^m and Q = 3ⁿ. Break the base 12 into its prime powers of 2 and 3, then rebuild each prime power out of P and Q using the exponent rules. A quick number test at the end pins down the single matching choice.
Factor the base into primes
Factoring the base gives two prime powers.
Splitting 12 into its 2-part and 3-part lets us handle the twos and threes separately.
8.EE.A.1Introduce A VariableRebuild the prime powers from P and Q
One prime appears twice, so its exponent doubles.
Stacking one exponent on another multiplies them, so P and Q can reach any exponent we want.
Stacking one exponent on another multiplies them, so the named quantities can reach any exponent needed.
▸ Why?
An exponent counts how many times a factor is used, so a power of a power uses it that many times over.
▸ Why?
Splitting the base into its primes lets the twos and the threes be handled entirely separately.
Match the pieces together
Matching the pieces gives P²ⁿQ^m.
Once each prime power is written with P or Q, the target is just their product.
6.EE.A.2Organize Information In More WaysConfirm with a quick number test
A quick numeric test confirms P²ⁿQ^m, choice (D).
Plugging in small numbers confirms the algebra and knocks out look-alike choices.
6.EE.A.1Eliminate PossibilitiesBreak a number into its prime powers, then rebuild the target one prime at a time using the exponent rules.
- Factor the base into primes
- Rebuild the prime powers from P and Q
- Match the pieces together
- Confirm with a quick number test