Let's analyze the expression given:

$\frac{e^{-3x}}{e^{-4x}}$

Step-by-step solution:

We know that when dividing two exponential expressions with the same base, we subtract the exponents:

$\frac{e^{-3x}}{e^{-4x}} = e^{-3x - (-4x)} = e^{-3x + 4x} = e^{x}$

Final Answer:

The expression simplifies to:

$e^{x}$

This matches the required form $e^{ax + b}$ where $a = 1$ and $b = 0$, with $a, b \in \mathbb{Z}$.

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

Here are five related questions to expand your understanding:

1. How do you simplify expressions with different exponential bases?
2. What is the rule for multiplying two exponential terms with the same base?
3. How do you handle exponents when dividing two numbers with the same base?
4. What is the general form of an exponential function?
5. How can you convert an expression in logarithmic form to exponential form?

Tip: Remember that dividing exponential terms with the same base always results in subtracting the exponents.