Simplify Radical Expression F232Ce

1. **State the problem:** Simplify the expression $$(\sqrt{10} - \sqrt{12})(12\sqrt{10} - 10\sqrt{12})$$. 2. **Use the distributive property (FOIL) to expand:** $$ (\sqrt{10})(12\sqrt{10}) - (\sqrt{10})(10\sqrt{12}) - (\sqrt{12})(12\sqrt{10}) + (\sqrt{12})(10\sqrt{12}) $$ 3. **Simplify each term:** - $$(\sqrt{10})(12\sqrt{10}) = 12(\sqrt{10} \times \sqrt{10}) = 12 \times 10 = 120$$ - $$(\sqrt{10})(10\sqrt{12}) = 10(\sqrt{10} \times \sqrt{12}) = 10\sqrt{120}$$ - $$(\sqrt{12})(12\sqrt{10}) = 12(\sqrt{12} \times \sqrt{10}) = 12\sqrt{120}$$ - $$(\sqrt{12})(10\sqrt{12}) = 10(\sqrt{12} \times \sqrt{12}) = 10 \times 12 = 120$$ 4. **Rewrite the expression with simplified terms:** $$ 120 - 10\sqrt{120} - 12\sqrt{120} + 120 $$ 5. **Combine like terms:** $$ 120 + 120 - (10\sqrt{120} + 12\sqrt{120}) = 240 - 22\sqrt{120} $$ 6. **Simplify the radical $\sqrt{120}$:** $$ \sqrt{120} = \sqrt{4 \times 30} = \sqrt{4} \times \sqrt{30} = 2\sqrt{30} $$ 7. **Substitute back:** $$ 240 - 22 \times 2 \sqrt{30} = 240 - 44\sqrt{30} $$ **Final answer:** $$ \boxed{240 - 44\sqrt{30}} $$