AMC 10 · 2009 · #13
Grade 8 arithmeticSuppose that P=2m and Q=3n. Which of the following is equal to 12mn for every pair of integers (m,n)?
Pick an answer.
AMC 10 2009 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: The letters $P$ and $Q$ stand for the numbers $P = 2^m$ and $Q = 3^n$. Among the five answer choices, find the one that is equal to $12^{mn}$ no matter which integers $m$ and $n$ are chosen.
Givens: $P = 2^m$; $Q = 3^n$; The target expression is $12^{mn}$; Choices: (A) $P^2Q$, (B) $P^nQ^m$, (C) $P^nQ^{2m}$, (D) $P^{2m}Q^n$, (E) $P^{2n}Q^m$
Unknowns: Which choice equals $12^{mn}$ for every pair of integers $(m,n)$
Understand
Restated: The letters $P$ and $Q$ stand for the numbers $P = 2^m$ and $Q = 3^n$. Among the five answer choices, find the one that is equal to $12^{mn}$ no matter which integers $m$ and $n$ are chosen.
Givens: $P = 2^m$; $Q = 3^n$; The target expression is $12^{mn}$; Choices: (A) $P^2Q$, (B) $P^nQ^m$, (C) $P^nQ^{2m}$, (D) $P^{2m}Q^n$, (E) $P^{2n}Q^m$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #15 Organize Information in More Ways, #3 Eliminate Possibilities
Everything is written with the two building blocks $P = 2^m$ and $Q = 3^n$. 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.
Execute — Answer: E
8.EE.A.1 Step 1 Factor the base into primes
- The base $12$ is $2^2 \times 3$.
- Raising a product to a power raises each factor, so the two prime pieces separate cleanly.
💡 Splitting $12$ into its $2$-part and $3$-part lets us handle the twos and threes separately.
8.EE.A.1 Step 2 Rebuild the prime powers from P and Q
- Raising a power to another power multiplies the exponents.
- So $P^{2n} = (2^m)^{2n} = 2^{2mn}$, and $Q^m = (3^n)^m = 3^{mn}$.
- These are exactly the two pieces we need.
💡 Stacking one exponent on another multiplies them, so $P$ and $Q$ can reach any exponent we want.
6.EE.A.2 Step 3 Match the pieces together
- The target broke into $2^{2mn} \cdot 3^{mn}$, and those are the same as $P^{2n}$ and $Q^m$.
- Multiply them back together to rebuild $12^{mn}$ entirely from $P$ and $Q$.
💡 Once each prime power is written with $P$ or $Q$, the target is just their product.
6.EE.A.1 Step 4 Confirm with a quick number test
- Try $m = 1$ and $n = 1$.
- Then $12^{mn} = 12$, while $P = 2$ and $Q = 3$, so $P^{2n}Q^m = 2^2 \cdot 3 = 12$.
- It matches, and the other choices give different values (for example $P^2Q = 12$ here too, but it fails when $m=2,n=1$: $12^2 = 144$ while $P^2Q = 4^2\cdot 3 = 48$).
- So the answer is (E).
💡 Plugging in small numbers confirms the algebra and knocks out look-alike choices.
8.EE.A.1 The base $12$ is $2^2 \times 3$. Raising a product to a power raises each factor 8.EE.A.1 Raising a power to another power multiplies the exponents. So $P^{2n} = (2^m)^{2 6.EE.A.2 The target broke into $2^{2mn} \cdot 3^{mn}$, and those are the same as $P^{2n}$ 6.EE.A.1 Try $m = 1$ and $n = 1$. Then $12^{mn} = 12$, while $P = 2$ and $Q = 3$, so $P^{ Review
Reasonableness: The base $12 = 2^2 \cdot 3$ carries twice as many factors of $2$ as of $3$, so the $2$-part should have the bigger exponent. Choice (E) uses $P^{2n}$ (the $2$-part) against $Q^m$ (the $3$-part), and $2n$ beats $m$ in exactly the doubled way $2^2\cdot 3$ predicts. Testing $m=2, n=1$ gives $144$ on both sides, confirming the match holds beyond a single lucky pair.
Alternative: Skip the letters and compare exponents directly. Write each choice back in terms of $2$ and $3$ and demand it equal $2^{2mn}3^{mn}$. Only (E) gives a $2$-exponent of $2mn$ and a $3$-exponent of $mn$ at the same time.
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Distributing the exponent over $2^2\cdot 3$ and multiplying stacked exponents to turn $P$ and $Q$ into the needed powers of $2$ and $3$.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Reassembling $12^{mn}$ out of the given letter-quantities $P$ and $Q$ once each prime power is identified.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating $12^{mn}$ and $P^{2n}Q^m$ at small integer values to confirm the match and eliminate other choices.)
⭐ Break a number into its prime powers, then rebuild the target one prime at a time using the exponent rules.
⭐ Break a number into its prime powers, then rebuild the target one prime at a time using the exponent rules.
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