1. **State the problem:** Simplify or analyze the expression $\left(\sqrt{x-2}\right) x^2$. 2. **Recall the formula and rules:** The square root function is defined as $\sqrt{a} = a^{\frac{1}{2}}$ for $a \geq 0$. Here, $\sqrt{x-2}$ means the square root of $x-2$, so the domain requires $x-2 \geq 0$, or $x \geq 2$. 3. **Rewrite the expression:** $$\left(\sqrt{x-2}\right) x^2 = (x-2)^{\frac{1}{2}} x^2$$ 4. **Interpretation:** This expression is the product of $x^2$ and the square root of $x-2$. It cannot be simplified further algebraically without additional context. 5. **Domain:** The expression is defined for all $x$ such that $x \geq 2$. **Final answer:** $$\boxed{(x-2)^{\frac{1}{2}} x^2}$$ This is the simplified form with domain $x \geq 2$.