Why does the distributive property work?

Answer 1

The distributive property works because multiplication can be expressed in terms of addition. For example, 3 * 4 = 3 + 3 + 3 + 3. So 3 * (2 + 4) = 3 + 3 + 3 + 3 + 3 + 3. Also, 3 * 2 + 3 * 4 = (3 + 3) + (3 + 3 + 3 + 3). Either way you have 6 3's added together or 18. Therefore, 3 * (2 + 4) = 3 * 2 + 3 * 4. This example can be generalized using summation notation to prove the distributive property for all integers.