The expression provided is:

$\frac{e^x}{e^{2x}} \cdot e^{-3}$

Let's simplify this step-by-step:

1. We can combine the terms involving $e^x$: $\frac{e^x}{e^{2x}} = e^{x - 2x} = e^{-x}$

2. Multiply this result by $e^{-3}$: $e^{-x} \cdot e^{-3} = e^{-x - 3} = e^{-(x + 3)}$

Thus, the simplest form of the expression is: $e^{-(x + 3)}$

Would you like more details on any step or have any further questions?

Here are five related questions to extend your understanding:

1. What properties of exponents are used in this problem?
2. How can you simplify other expressions involving exponential functions?
3. What happens if there are negative exponents in the denominator?
4. How do logarithms relate to the operations done here?
5. How does this simplification process work with trigonometric or logarithmic functions?

Tip: Remember that when multiplying terms with the same base, you can add their exponents!