<s>[INST] I can't find 'the easiest way' to simplify this expression? $\large\dfrac{3^{2008} (10^{2013} + 5^{2012} + 2^{2011})}{5^{2012} (6^{2010} + 3^{2009} + 2^{2008})}$ 
$\large=\dfrac{3^{2008} ((2\cdot5)^{2013} + 5^{2012} + 2^{2011})}{5^{2012} ((2\cdot3)^{2010} + 3^{2009} + 2^{2008})}$   
$\large=\dfrac{3^{2008} (2^{2013}\cdot5^{2013} + 5^{2012} + 2^{2011})}{5^{2012} (2^{2010}\cdot3^{2010} + 3^{2009} + 2^{2008})}$  
This is where I get stuck. Any help would be appreciated.  
 [/INST] The expression is extremely close to $\dfrac{40}{9}$. This is because in the numerator, $2^{2013}\cdot 5^{2013}$ is much larger than the remaining terms $5^{2012}$ and $2^{2011}$, so the remaining terms are almost negligible. Similarly for the denominator. The expression then simplifies to$\dfrac{2^{2013}\cdot 3^{2008}\cdot 5^{2013}}{2^{2010}\cdot 3^{2010}\cdot 5^{2012}}=\dfrac{40}{9}$.
I doubt there is a simpler expression, since the numerator is divisible by $3^{2008}$ but the denominator is not divisible by $3$, while the denominator is divisible by $5^{2012}$ but the numerator is not divisible by $5$.
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