1. **State the problem:** Simplify the expressions $\sqrt{112}$, $\sqrt{63}$, and then find the sum $\sqrt{112} + \sqrt{63}$.\n\n2. **Recall the formula and rules:** The square root of a product can be written as the product of square roots: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$\nWe look for perfect squares inside the radicand to simplify the square root.\n\n3. **Simplify $\sqrt{112}$:**\nFactor 112 into $16 \times 7$ because 16 is a perfect square.\n$$\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}$$\n\n4. **Simplify $\sqrt{63}$:**\nFactor 63 into $9 \times 7$ because 9 is a perfect square.\n$$\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}$$\n\n5. **Add the simplified square roots:**\n$$\sqrt{112} + \sqrt{63} = 4\sqrt{7} + 3\sqrt{7} = (4 + 3)\sqrt{7} = 7\sqrt{7}$$\n\n**Final answers:**\n$$\sqrt{112} = 4\sqrt{7}$$\n$$\sqrt{63} = 3\sqrt{7}$$\n$$\sqrt{112} + \sqrt{63} = 7\sqrt{7}$$