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Difference between revisions of "2013 AMC 10A Problems/Problem 8"
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==Problem==
 
==Problem==
  
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What is the value of <math>\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?</math>  
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What is the value of <cmath>\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?</cmath>  
  
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<math> \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1  \qquad\textbf{(C)}\ \frac{5}{3} \qquad\textbf{(D)}\ 2013 \qquad\textbf{(E)}\ 2^{4024} </math>
  
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==Solution==
  
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<math> \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{5}{3} \qquad\textbf{(D)}\ 2013 \qquad\textbf{(E)}\ 2^{4024} </math>
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Factoring out, we get: <math>\frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2-1)}</math>.  
 +
 
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Cancelling out the <math>2^{2012}</math> from the numerator and denominator, we see that it simplifies to <math>\boxed{\textbf{(C) }\frac{5}{3}}</math>.
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==Solution 2==
 +
Let <math>x=2^{2012}</math>
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Then the given expression is equal to <math>\frac{4x+x}{4x-x}=\frac{5x}{3x}=\boxed{\textbf{(C) }\frac{5}{3}}</math>
  
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==Video Solution (CREATIVE THINKING)==
 +
https://youtu.be/1ULw0x9XK4o
  
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==Solution==
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~Education, the Study of Everything
  
  
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=2vf843cvVzo?t=545
  
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Factoring out, we get: <math>\frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2-1)} ?</math>
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~sugar_rush
  
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https://youtu.be/C3prgokOdHc
  
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Cancelling out the <math>2^{2012}</math> from the numerator and denominator, we see that it simplifies to <math>\boxed{\textbf{(C) }\frac{5}{3}}</math>.
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~savannahsolver
  
 
==See Also==
 
==See Also==
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{{AMC10 box|year=2013|ab=A|num-b=7|num-a=9}}
 
{{AMC10 box|year=2013|ab=A|num-b=7|num-a=9}}
 
{{AMC12 box|year=2013|ab=A|num-b=3|num-a=5}}
 
{{AMC12 box|year=2013|ab=A|num-b=3|num-a=5}}
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{{MAA Notice}}

Latest revision as of 12:08, 1 July 2023

Problem

What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?\]

$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{5}{3} \qquad\textbf{(D)}\ 2013 \qquad\textbf{(E)}\ 2^{4024}$

Solution

Factoring out, we get: $\frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2-1)}$.

Cancelling out the $2^{2012}$ from the numerator and denominator, we see that it simplifies to $\boxed{\textbf{(C) }\frac{5}{3}}$.

Solution 2

Let $x=2^{2012}$

Then the given expression is equal to $\frac{4x+x}{4x-x}=\frac{5x}{3x}=\boxed{\textbf{(C) }\frac{5}{3}}$

Video Solution (CREATIVE THINKING)

https://youtu.be/1ULw0x9XK4o

~Education, the Study of Everything


Video Solution

https://www.youtube.com/watch?v=2vf843cvVzo?t=545

~sugar_rush

https://youtu.be/C3prgokOdHc

~savannahsolver

See Also

2013 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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
2013 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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

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