Radical Simplification 6999C3

1. **Problem:** Expand and simplify $\sqrt{3}(6\sqrt{3} + 2)$. 2. **Formula:** Use distributive property: $a(b+c) = ab + ac$. 3. **Step 1:** Multiply $\sqrt{3}$ by each term inside the parentheses: $$\sqrt{3} \times 6\sqrt{3} + \sqrt{3} \times 2 = 6\sqrt{3} \times \sqrt{3} + 2\sqrt{3}$$ 4. **Step 2:** Simplify $\sqrt{3} \times \sqrt{3} = \sqrt{3 \times 3} = \sqrt{9} = 3$: $$6 \times 3 + 2\sqrt{3} = 18 + 2\sqrt{3}$$ 5. **Final answer:** $18 + 2\sqrt{3}$. --- 1. **Problem:** Simplify $6\sqrt{3} + \sqrt{75}$. 2. **Step 1:** Simplify $\sqrt{75}$: $$\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$$ 3. **Step 2:** Substitute back: $$6\sqrt{3} + 5\sqrt{3}$$ 4. **Step 3:** Combine like terms: $$6\sqrt{3} + 5\sqrt{3} = (6 + 5)\sqrt{3} = 11\sqrt{3}$$ 5. **Final answer:** $11\sqrt{3}$. --- 1. **Problem:** Simplify $\sqrt{11} \times \sqrt{22}$. 2. **Step 1:** Use property $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$: $$\sqrt{11} \times \sqrt{22} = \sqrt{11 \times 22} = \sqrt{242}$$ 3. **Step 2:** Factor $242$: $$242 = 121 \times 2$$ 4. **Step 3:** Simplify: $$\sqrt{242} = \sqrt{121 \times 2} = \sqrt{121} \times \sqrt{2} = 11\sqrt{2}$$ 5. **Final answer:** $11\sqrt{2}$. --- 1. **Problem:** Simplify $5\sqrt{4} \times 2\sqrt{28}$. 2. **Step 1:** Multiply coefficients and radicals separately: $$5 \times 2 \times \sqrt{4} \times \sqrt{28} = 10 \times \sqrt{4 \times 28}$$ 3. **Step 2:** Simplify inside the radical: $$4 \times 28 = 112$$ 4. **Step 3:** Simplify $\sqrt{112}$: $$\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}$$ 5. **Step 4:** Substitute back: $$10 \times 4\sqrt{7} = 40\sqrt{7}$$ 6. **Final answer:** $40\sqrt{7}$.