1. **State the problem:** Find the product of the radicals in the expression $3\sqrt{2}(5\sqrt{12} - \sqrt{18} + 4\sqrt{24})$. 2. **Recall the formula and rules:** - Multiplying radicals: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. - Simplify radicals by factoring out perfect squares. - Distribute multiplication over addition/subtraction. 3. **Simplify each radical inside the parentheses:** - $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ - $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ - $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ 4. **Rewrite the expression:** $$3\sqrt{2} \left(5 \times 2\sqrt{3} - 3\sqrt{2} + 4 \times 2\sqrt{6}\right) = 3\sqrt{2} (10\sqrt{3} - 3\sqrt{2} + 8\sqrt{6})$$ 5. **Distribute $3\sqrt{2}$ to each term:** - $3\sqrt{2} \times 10\sqrt{3} = 30 \sqrt{2 \times 3} = 30\sqrt{6}$ - $3\sqrt{2} \times (-3\sqrt{2}) = -9 \sqrt{2 \times 2} = -9 \times 2 = -18$ - $3\sqrt{2} \times 8\sqrt{6} = 24 \sqrt{2 \times 6} = 24\sqrt{12}$ 6. **Simplify $\sqrt{12}$ in the last term:** - $\sqrt{12} = 2\sqrt{3}$ - So, $24\sqrt{12} = 24 \times 2\sqrt{3} = 48\sqrt{3}$ 7. **Combine all terms:** $$30\sqrt{6} - 18 + 48\sqrt{3}$$ **Final answer:** $30\sqrt{6} - 18 + 48\sqrt{3}$