1. **State the problem:** Simplify or analyze the expression $$\frac{(a - 3)^{1 - \sqrt{2}}}{a + 2}$$ where $a$ is a variable. 2. **Recall the rules:** - When you have an expression with an exponent like $x^m$, it means $x$ multiplied by itself $m$ times. - The exponent here is $1 - \sqrt{2}$, which is an irrational number since $\sqrt{2}$ is irrational. - The denominator is $a + 2$, which cannot be zero (so $a \neq -2$). 3. **Rewrite the expression:** $$\frac{(a - 3)^{1 - \sqrt{2}}}{a + 2} = \frac{(a - 3)^1}{(a - 3)^{\sqrt{2}}(a + 2)} = \frac{a - 3}{(a - 3)^{\sqrt{2}}(a + 2)}$$ 4. **Simplify the numerator and denominator:** $$= \frac{a - 3}{(a - 3)^{\sqrt{2}}(a + 2)} = \frac{\cancel{a - 3}}{(a - 3)^{\sqrt{2}}(a + 2)} \text{ only if } a - 3 \neq 0$$ Since $a - 3$ in numerator and denominator are not the same power, we cannot cancel directly. So the expression remains as is. 5. **Final expression:** $$\frac{(a - 3)^{1 - \sqrt{2}}}{a + 2}$$ 6. **Domain restrictions:** - $a \neq -2$ (denominator zero) - $a - 3 > 0$ if we want real values for the irrational exponent (since fractional powers of negative numbers are not real). **Answer:** The expression is $$\frac{(a - 3)^{1 - \sqrt{2}}}{a + 2}$$ with domain $a > 3$ and $a \neq -2$.