We are asked to simplify the expression $(\frac{a-3}{a} - \frac{a-3}{a^2 - 2a})^{a-2} : \frac{a-3}{a}$.

1. Problem Description

We are asked to simplify the expression $(\frac{a-3}{a} - \frac{a-3}{a^2 - 2a})^{a-2} : \frac{a-3}{a}$.

2. Solution Steps

First, let's simplify the expression inside the parenthesis.

$\frac{a-3}{a} - \frac{a-3}{a^2 - 2a}$

Factor the denominator of the second fraction: $a^2 - 2a = a(a-2)$.

$\frac{a-3}{a} - \frac{a-3}{a(a-2)}$

Find a common denominator, which is $a(a-2)$.

$\frac{(a-3)(a-2)}{a(a-2)} - \frac{a-3}{a(a-2)}$

Combine the fractions:

$\frac{(a-3)(a-2) - (a-3)}{a(a-2)}$

Factor out $(a-3)$ from the numerator:

$\frac{(a-3)(a-2-1)}{a(a-2)}$

$\frac{(a-3)(a-3)}{a(a-2)}$

$\frac{(a-3)^2}{a(a-2)}$

Now, consider the entire expression:

$(\frac{(a-3)^2}{a(a-2)})^{a-2} : \frac{a-3}{a}$

$(\frac{(a-3)^2}{a(a-2)})^{a-2} \cdot \frac{a}{a-3}$

$\frac{(a-3)^{2(a-2)}}{a^{a-2}(a-2)^{a-2}} \cdot \frac{a}{a-3}$

$\frac{(a-3)^{2a-4}}{a^{a-2}(a-2)^{a-2}} \cdot \frac{a}{a-3}$

$\frac{(a-3)^{2a-4} a}{a^{a-2} (a-2)^{a-2} (a-3)}$

$\frac{(a-3)^{2a-5} a}{a^{a-2} (a-2)^{a-2}}$

$\frac{(a-3)^{2a-5}}{a^{a-3} (a-2)^{a-2}}$

3. Final Answer

$\frac{(a-3)^{2a-5}}{a^{a-3}(a-2)^{a-2}}$