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

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