Determine the following: \( a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} \)

To simplify the expression \( a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} \), we can use the properties of exponents. Specifically, we can combine the terms with the same base by adding their exponents. First, we will group the terms with base \( a \) and the terms with base \( b \): 1. For the base \( a \): \[ a^{4} \cdot a^{2} \cdot a^{-5} = a^{4 + 2 - 5} = a^{1} = a \] 2. For the base \( b \): \[ b^{-9} \cdot b^{7} \cdot b^{2} = b^{-9 + 7 + 2} = b^{0} \] Since \( b^{0} = 1 \), we can now combine the results: \[ a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} = a \cdot 1 = a \] Thus, the simplified expression is: \[ \boxed{a} \]