Can you help me understand why and how the distributive law in mathematics works?
Certainly! The distributive law is a fundamental property of arithmetic that states that multiplying a number by a sum of two or more numbers is the same as multiplying each addend separately and then adding the products. This can be expressed as follows:

a × (b + c) = (a × b) + (a × c)

where a, b, and c are any real numbers.

To understand why the distributive law works, let's consider a concrete example:

3 × (4 + 5)

To evaluate this expression, we can apply the distributive law as follows:

3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27

We can see that we get the same result as if we added 4 and 5 first and then multiplied the sum by 3:

3 × (4 + 5) = 3 × 9 = 27

The distributive law works because multiplication is distributive over addition, meaning that multiplying a number by a sum is the same as multiplying each addend separately and then adding the products. This property allows us to simplify expressions and perform calculations more efficiently.

The distributive law is also a foundational concept in algebra, where it is used to simplify expressions, factor polynomials, and solve equations. For example, we can use the distributive law to factor the expression 6x + 9y as follows:

6x + 9y = 3(2x + 3y)

In this case, we can see that the distributive law allows us to factor out the greatest common factor of 3 from both terms, which simplifies the expression and makes it easier to work with.

I hope that helps you understand why and how the distributive law works!