The distributive property of multiplication states that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers separately and then adding the products.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra and arithmetic that allows you to simplify expressions. It essentially "distributes" a factor across terms within parentheses.
The Formula
The distributive property can be represented by the following formula:
a (b + c) = (a b) + (a * c)
Where 'a', 'b', and 'c' can be any real numbers.
Examples
Let's illustrate with some examples:
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Example 1: 3 * (2 + 4)
- Using the distributive property: (3 2) + (3 4) = 6 + 12 = 18
- Directly calculating: 3 * (6) = 18
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Example 2: 5 * (x + 2)
- Using the distributive property: (5 x) + (5 2) = 5x + 10
Applications
The distributive property is widely used in:
- Algebraic simplification: As seen in Example 2, it's used to simplify expressions with variables.
- Mental math: It can make calculations easier. For example, 6 102 can be thought of as 6 (100 + 2) = (6 100) + (6 2) = 600 + 12 = 612.
- Factoring: The distributive property can also be used in reverse to factor expressions.
Distributive Property with Subtraction
The distributive property also applies to subtraction:
a (b - c) = (a b) - (a * c)
For example:
- 4 (5 - 2) = (4 5) - (4 * 2) = 20 - 8 = 12
Summary
In summary, the distributive property allows you to multiply a single term by two or more terms inside a set of parentheses by multiplying the outside term by each of the inside terms. This property holds true for both addition and subtraction.
