\( \frac { \sqrt { 10 ^ { 2009 } } } { 10 ^ { 2011 } \cdot \sqrt { 10 ^ { 2007 } } } \quad \frac { 5 ^ { 2007 } + 5 ^ { 2010 } } { 5 ^ { 2008 } + 5 ^ { 2009 } } \)

Let's simplify the expression step by step. 1. **First expression:** We start with the first part, which is \( \frac { \sqrt { 10 ^ { 2009 } } } { 10 ^ { 2011 } \cdot \sqrt { 10 ^ { 2007 } } } \). We can simplify \( \sqrt{10^{2009}} = 10^{2009/2} = 10^{1004.5} \) and \( \sqrt{10^{2007}} = 10^{2007/2} = 10^{1003.5} \). Thus, the denominator becomes: \[ 10^{2011} \cdot \sqrt{10^{2007}} = 10^{2011} \cdot 10^{1003.5} = 10^{2011 + 1003.5} = 10^{3014.5}. \] Now, we have: \[ \frac{10^{1004.5}}{10^{3014.5}} = 10^{1004.5 - 3014.5} = 10^{-2010}. \] 2. **Second expression:** Next, we simplify the second part, which is \( \frac { 5 ^ { 2007 } + 5 ^ { 2010 } } { 5 ^ { 2008 } + 5 ^ { 2009 } } \). Factoring out \( 5^{2007} \) from the numerator: \[ 5^{2007}(1 + 5^{3}) = 5^{2007}(1 + 125) = 5^{2007} \cdot 126. \] Now factor out \( 5^{2008} \) from the denominator: \[ 5^{2008}(1 + 5^{1}) = 5^{2008}(1 + 5) = 5^{2008} \cdot 6. \] This means we can now rewrite the expression as: \[ \frac{5^{2007} \cdot 126}{5^{2008} \cdot 6} = \frac{126}{6} = 21. \] Putting everything together, our final expression is: \[ 10^{-2010} \cdot 21. \] Hence, the final answer is: \[ 21 \times 10^{-2010}. \]