1. **State the problem:** Simplify the radical expression $\sqrt{486a^4b}$.\n\n2. **Recall the formula and rules:** The square root of a product is the product of the square roots: $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$\nAlso, for even powers inside a square root, $$\sqrt{a^{2n}} = a^n$$ if $a \geq 0$.\n\n3. **Factor the radicand:**\n$$486a^4b = 81 \times 6 \times a^4 \times b$$\nSince $81 = 9^2$, we can write:\n$$\sqrt{486a^4b} = \sqrt{9^2 \times 6 \times a^4 \times b}$$\n\n4. **Apply the square root to perfect squares:**\n$$\sqrt{9^2} = 9$$\n$$\sqrt{a^4} = a^{\cancel{4}/2} = a^2$$\nIntermediate step showing cancellation:\n$$\sqrt{a^{\cancel{4}}} = a^{\cancel{4}/2} = a^2$$\n\n5. **Rewrite the expression:**\n$$\sqrt{486a^4b} = 9a^2 \sqrt{6b}$$\n\n6. **Final answer:**\n$$\boxed{9a^2 \sqrt{6b}}$$