Can you help me understand why and how the distributive law in mathematics works?
user
Certainly! The distributive law is a fundamental property of mathematics that allows us to simplify expressions involving multiplication and addition (or subtraction). The law states that:

a × (b + c) = a × b + a × c

In words, this means that when you multiply a number (a) by the sum of two other numbers (b and c), you can distribute the multiplication over the addition, and get the same result as if you had multiplied a by b and then added the result to a times c.

The distributive law works because multiplication is distributive over addition. In other words, if you have two numbers (a and b) and you add them together, and then multiply the result by a third number (c), you can either multiply a by c and add the result to b times c, or you can multiply the sum of a and b by c, and you will get the same result.

To see why this works, let's use some specific numbers. Suppose we want to simplify the expression:

3 × (4 + 5)

Using the distributive law, we can rewrite this as:

3 × 4 + 3 × 5

Which equals:

12 + 15 = 27

We could also have simply multiplied 3 by the sum of 4 and 5 directly, without using the distributive law:

3 × (4 + 5) = 3 × 9 = 27

Either way, we get the same result.

The distributive law is a very useful tool in algebra and other areas of mathematics, because it allows us to simplify complex expressions and equations, and to work with variables and unknowns. For example, we can use the distributive law to expand expressions like:

3(x + 2)

Which equals:

3x + 6

I hope that helps! Let me know if you have any further questions.
assistant