1. **Problem:** Simplify the expression $5\sqrt{7} \times (3\sqrt{2} + 7\sqrt{7})$. 2. **Formula and rules:** - Use distributive property: $a(b+c) = ab + ac$. - Multiply radicals: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. - Simplify square roots if possible. 3. **Step-by-step solution:** 1. Distribute $5\sqrt{7}$: $$5\sqrt{7} \times 3\sqrt{2} + 5\sqrt{7} \times 7\sqrt{7}$$ 2. Multiply coefficients and radicals: $$15 \times \sqrt{7 \times 2} + 35 \times \sqrt{7 \times 7}$$ 3. Simplify inside radicals: $$15 \sqrt{14} + 35 \sqrt{49}$$ 4. Since $\sqrt{49} = 7$: $$15 \sqrt{14} + 35 \times 7$$ 5. Multiply: $$15 \sqrt{14} + 245$$ **Final answer:** $$15 \sqrt{14} + 245$$