Difference between revisions of "2013 AMC 10A Problems/Problem 8"

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==Problem==
 
==Problem==
  
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>  
  
 
<math> \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1  \qquad\textbf{(C)}\ \frac{5}{3} \qquad\textbf{(D)}\ 2013 \qquad\textbf{(E)}\ 2^{4024} </math>
 
<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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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>.
 
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==
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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>
  
 
==See Also==
 
==See Also==

Revision as of 22:22, 23 November 2020

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 $\frac{4x+x}{4x-x}=\frac{5x}{3x}=\boxed{\textbf{(C)}\frac{5}{3}}$

See Also

2013 AMC 10A (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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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