Distributive Law of Multiplication: Simplifying Arithmetic and Algebra

Understand the distributive law of multiplication and how it simplifies calculations involving multiplication and addition or subtraction. This guide provides a clear explanation, along with illustrative examples, showing how to apply the distributive law in arithmetic and algebraic expressions.

Distributive Law of Multiplication: Simplifying Arithmetic and Algebra

Introduction to the Distributive Law

The distributive law is a fundamental concept in mathematics that simplifies calculations involving multiplication and addition or subtraction. It states that multiplying a number by a sum or difference is the same as multiplying the number by each term in the sum or difference and then adding or subtracting the results. This is a very useful property for simplifying mathematical expressions.

The Distributive Law Formula

The distributive law is expressed as:

x * (y + z) = x*y + x*z

This means multiplying x by the sum of y and z is the same as multiplying x by y and then adding the result of multiplying x by z.

Example: Applying the Distributive Law

Let's illustrate with a simple example: 9 * (5 + 1).

  1. Distribute the 9: 9 * 5 + 9 * 1
  2. Calculate: 45 + 9 = 54

The result is the same as calculating 9 * (5 + 1) = 9 * 6 = 54.

Distributive Law with Variables

The distributive law also applies to algebraic expressions containing variables. For example:

5 * (6 + 8x) = 5 * 6 + 5 * 8x = 30 + 40x

Distributive Law Over Addition and Subtraction

1. Distributive Law Over Addition:

Multiplying a number by a sum:

7 * (9 + 4) = 7 * 9 + 7 * 4 = 63 + 28 = 91

2. Distributive Law Over Subtraction:

Multiplying a number by a difference:

7 * (9 - 4) = 7 * 9 - 7 * 4 = 63 - 28 = 35

Distributive Law and Division

The distributive law can also be applied to division, but it only works when dividing a sum by a single number:

(60 + 24) / 6 = (60 / 6) + (24 / 6) = 10 + 4 = 14

Examples of the Distributive Law in Algebra

4 * (2x + 7x) = 8x + 28x = 36x
4 * (8xy + 12yx) = 32xy + 48xy = 80xy

Why the Distributive Law Doesn't Apply to Multiplication and Division

The distributive law does *not* apply to multiplication over multiplication or division. For example:

5 * (4 * 8) ≠ (5 * 4) * (5 * 8)  (160 ≠ 800)
5 * (9 / 3) ≠ (5 * 9) / (5 * 3)  (15 ≠ 3)

Conclusion

The distributive law is a valuable tool for simplifying mathematical expressions, particularly in algebra. Understanding its applications and limitations is essential for anyone working with mathematics.