We want to prove that:

$e^{x - y} = \frac{e^x}{e^y}$

Proof:

The left-hand side of the equation is $e^{x - y}$. Using the exponentiation rule that says:

$a^{m - n} = \frac{a^m}{a^n}$

for any base $a$, we can apply this directly to $e$, which is the base of natural logarithms. So, we have:

$e^{x - y} = \frac{e^x}{e^y}$

Thus, the expression on the left-hand side is exactly equal to the expression on the right-hand side.

Conclusion:

The identity is true by applying the exponentiation rule.

Do you need further details or have any questions?

Here are some related questions:

1. What is the general rule for dividing exponential functions?
2. Can this rule be applied to logarithms as well?
3. How does this property apply when working with derivatives of exponential functions?
4. What other properties of exponentials can be derived from this?
5. How can this rule be used in solving differential equations?

Tip: When working with exponents, remember that subtraction in the exponent translates to division of the same base.