Difference between revisions of "2009 AMC 12A Problems/Problem 6"

(New page: == Problem == Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>? <math>\tex...)
 
 
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{{duplicate|[[2009 AMC 12A Problems|2009 AMC 12A #6]] and [[2009 AMC 10A Problems|2009 AMC 10A #13]]}}
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== Problem ==
 
== Problem ==
 
Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>?
 
Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>?
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== Solution ==
 
== Solution ==
  
We have <math>12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{P^{2n} Q^m}</math>.
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We have <math>12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{\bold{E)} P^{2n} Q^m}</math>.
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==Video Solution==
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https://youtu.be/T3XXMO3YvHQ
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~savannahsolver
  
 
== See Also ==
 
== See Also ==
  
 
{{AMC12 box|year=2009|ab=A|num-b=5|num-a=7}}
 
{{AMC12 box|year=2009|ab=A|num-b=5|num-a=7}}
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{{AMC10 box|year=2009|ab=A|num-b=12|num-a=14}}
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{{MAA Notice}}

Latest revision as of 18:25, 7 August 2020

The following problem is from both the 2009 AMC 12A #6 and 2009 AMC 10A #13, so both problems redirect to this page.

Problem

Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?

$\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m$

Solution

We have $12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{\bold{E)} P^{2n} Q^m}$.

Video Solution

https://youtu.be/T3XXMO3YvHQ

~savannahsolver

See Also

2009 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
2009 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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