Given two functions \( f(x)=4 \cdot 2^{x} \) and \( g(x)=4 \cdot 5^{x} \), find and simplify the following: a) \( (f g)(x)=\square \) b) \( \left(\frac{f}{g}\right)(x)=\square \)

To find the products and quotients of the functions \( f(x) \) and \( g(x) \), let's start with each calculation. a) The product of \( f(x) \) and \( g(x) \) is: \[ (f g)(x) = f(x) \cdot g(x) = (4 \cdot 2^{x}) \cdot (4 \cdot 5^{x}) = 16 \cdot 2^{x} \cdot 5^{x} \] We can combine \( 2^x \cdot 5^x \) into a single exponential term: \[ = 16 \cdot (2 \cdot 5)^{x} = 16 \cdot 10^{x} \] So, \[ (f g)(x) = 16 \cdot 10^{x} \] b) The quotient of \( f(x) \) by \( g(x) \) is: \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4 \cdot 2^{x}}{4 \cdot 5^{x}} = \frac{2^{x}}{5^{x}} \] This can be simplified as: \[ = \left(\frac{2}{5}\right)^{x} \] Thus, the final results are: \[ (f g)(x) = 16 \cdot 10^{x} \] \[ \left(\frac{f}{g}\right)(x) = \left(\frac{2}{5}\right)^{x} \]