1. **State the problem:** Simplify or understand the expression $$\left(\frac{a - 3}{a + 2}\right)^{1 - \sqrt{2}}$$. 2. **Recall the exponent rule:** For any base $x$ and exponents $m$ and $n$, $$x^{m-n} = \frac{x^m}{x^n}$$. 3. **Apply the exponent rule:** Rewrite the expression as $$\left(\frac{a - 3}{a + 2}\right)^1 \cdot \left(\frac{a - 3}{a + 2}\right)^{-\sqrt{2}} = \frac{a - 3}{a + 2} \cdot \left(\frac{a - 3}{a + 2}\right)^{-\sqrt{2}}$$. 4. **Use the negative exponent rule:** For any $x$, $$x^{-k} = \frac{1}{x^k}$$. 5. **Rewrite the second term:** $$\left(\frac{a - 3}{a + 2}\right)^{-\sqrt{2}} = \frac{1}{\left(\frac{a - 3}{a + 2}\right)^{\sqrt{2}}} = \left(\frac{a + 2}{a - 3}\right)^{\sqrt{2}}$$. 6. **Combine the terms:** $$\frac{a - 3}{a + 2} \cdot \left(\frac{a + 2}{a - 3}\right)^{\sqrt{2}} = \frac{a - 3}{a + 2} \cdot \frac{(a + 2)^{\sqrt{2}}}{(a - 3)^{\sqrt{2}}} = \frac{(a - 3)^{1}}{(a + 2)^{1}} \cdot \frac{(a + 2)^{\sqrt{2}}}{(a - 3)^{\sqrt{2}}}$$. 7. **Combine powers with the same base:** $$= \frac{(a - 3)^{1}}{(a - 3)^{\sqrt{2}}} \cdot \frac{(a + 2)^{\sqrt{2}}}{(a + 2)^{1}} = (a - 3)^{1 - \sqrt{2}} \cdot (a + 2)^{\sqrt{2} - 1}$$. 8. **Final simplified form:** $$\boxed{(a - 3)^{1 - \sqrt{2}} (a + 2)^{\sqrt{2} - 1}}$$. This shows the expression rewritten as a product of powers with positive exponents where possible, which can be easier to interpret or use in further calculations.