Distributive Property Formula

Introduction to Distributive Property Formula

The distributive property is a fundamental property in mathematics that applies to multiplication over addition or subtraction. It states that for any real numbers (or any elements of a mathematical system where addition and multiplication are defined), the product of a number with the sum or difference of two other numbers is equal to the sum or difference of the products of that number with each of the other numbers.

Distributive Property Formula

The distributive property can be expressed as follows:

For any real numbers a, b, and c:

Distributive property with addition:

a * (b + c) = (a * b) + (a * c)

This formula states that when you multiply a number (a) by the sum (b + c), it is equivalent to multiplying the number (a) by each term separately (b and c) and then adding the products together.

Distributive property with subtraction:

a * (b – c) = (a * b) – (a * c)

This formula states that when you multiply a number (a) by the difference (b – c), it is equivalent to multiplying the number (a) by each term separately (b and c) and then subtracting the products.

These formulas are known as the distributive property of multiplication over addition and subtraction, respectively. They are widely used in algebraic manipulations and simplifications, allowing you to expand or factor expressions involving multiplication and addition or subtraction.

Solved Examples on Distributive Property Formula

Example 1: Simplify the expression: 3 * (2 + 4)

Solution:

Using the distributive property, we can distribute the 3 to both terms inside the parentheses:

3 * (2 + 4) = (3 * 2) + (3 * 4) = 6 + 12 = 18

Therefore, 3 * (2 + 4) simplifies to 18.

Example 2: Simplify the expression: 5 * (7 – 3)

Solution:

Applying the distributive property, we distribute the 5 to both terms inside the parentheses:

5 * (7 – 3) = (5 * 7) – (5 * 3) = 35 – 15 = 20

Therefore, 5 * (7 – 3) simplifies to 20.

Example 3: Simplify the expression: 2 * (x + 3)

Solution:

Using the distributive property, we distribute the 2 to both terms inside the parentheses:

2 * (x + 3) = (2 * x) + (2 * 3) = 2x + 6

Therefore, 2 * (x + 3) simplifies to 2x + 6.

Frequently Asked Questions on Distributive Property Formula

1: What is the purpose of the distributive property?

Answer: The distributive property is used to simplify and manipulate algebraic expressions involving multiplication and addition or subtraction. It allows us to expand or factor expressions, making calculations and simplifications easier.

2: Can the distributive property be applied to division?

Answer: No, the distributive property only applies to multiplication over addition or subtraction. It does not apply to division. For division, different properties and rules, such as the quotient rule, are used.

3: Can the distributive property be applied to more than two terms inside parentheses?

Answer: Yes, the distributive property can be extended to more than two terms inside parentheses. For example, if you have three terms, a, b, and c, inside the parentheses, you can distribute a number, x, to all three terms as follows:

x * (a + b + c) = (x * a) + (x * b) + (x * c)

You distribute x to each term separately and add the products together.

4: Can the distributive property be used with variables?

Answer: Yes, the distributive property applies to both numbers and variables. You can use the distributive property with variables just as you would with numbers. The key is to distribute the coefficient or number to each term inside the parentheses.

5: Can the distributive property be applied to subtraction?

Answer: Yes, the distributive property can be applied to subtraction as well. When you have a term being subtracted inside the parentheses, you distribute the number or coefficient to each term separately and then subtract the products. The formula for the distributive property with subtraction is:

a * (b – c) = (a * b) – (a * c)

6: How Does the Distributive Property Work?

Answer: When we use the distributive property formula, we multiply the outside term with the terms inside the brackets and then add the terms to get the solution. For example, let us solve 15(4 + 3). First, we will multiply 15 with 4, then multiply 15 with 3, and then add the products to get the answer. This means 15 × (4 + 3) = (15 × 4) + (15 × 3) = 60 + 45 = 105.

7: What is the Distributive Property for Rational Numbers?

Answer: The distributive property states, if p, q, and r are three rational numbers, then the relation between the three is given as, p × (q + r) = (p × q) + (p × r). For example, 1/3(1/2 + 1/5) = (1/3 × 1/2) + (1/3 × 1/5) = 7/30.

8: How to Use Distributive Property with Fractions?

Answer: The distributive property is applicable to fractions in a similar way as it is used for numbers and variables. For example, let us solve the expression, 1/3(2/6 + 4/6) using the distributive property. We will first multiply 1/3 with 2/6 and then with 4/6. This means, 1/3(2/6 + 4/6) ⇒ (1/3 × 2/6) + (1/3 × 4/6) = 2/18 + 4/18 = 6/18 = 1/3.