You're on a roll — Question 8: Calculate the product: $7\sqrt{6}(2\sqrt{10}-5\sqrt{2})$ Step 1: Distribute $7\sqrt{6}$ to each term inside the parentheses. $$ 7\sqrt{6}(2\sqrt{10}) - 7\sqrt{6}(5\sqrt{2}) $$ Step 2: Multiply the coefficients and the radicals for the first term. $$ (7 \times 2)\sqrt{6 \times 10} = 14\sqrt{60} $$ Step 3: Multiply the coefficients and the radicals for the second term. $$ (7 \times 5)\sqrt{6 \times 2} = 35\sqrt{12} $$ So the expression becomes: $$ 14\sqrt{60} - 35\sqrt{12} $$ Step 4: Simplify the radicals $\sqrt{60}$ and $\sqrt{12}$. For $\sqrt{60}$: $$ \sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15} $$ For $\sqrt{12}$: $$ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} $$ Step 5: Substitute the simplified radicals back into the expression. $$ 14(2\sqrt{15}) - 35(2\sqrt{3}) $$ $$ 28\sqrt{15} - 70\sqrt{3} $$ The terms $28\sqrt{15}$ and $70\sqrt{3}$ are not like terms because their radical parts are different, so they cannot be combined further. The simplified product is $28\sqrt{15} - 70\sqrt{3}$. $$\boxed{28\sqrt{15} - 70\sqrt{3}}$$ What's next?