Distributive Property

The distributive property is a property of multiplication which helps to simplify expressions. It allows us to multiply a sum by multiplying each addend separately and then adding the products.

The distributive property can be expressed as:

a × (b + c) = a × b + a × c

This means that we can multiply a number (a) by the sum of two other numbers (b + c) by multiplying a by both b and c separately and then adding the products.

Example 1: Find the value of 7 × 52

We can use the distributive property to simplify this expression as follows:

7 × 52 = 7 × (50 + 2) (52 can be expressed as the sum of 50 and 2)
= 7 × 50 + 7 × 2 (multiply each addend separately)
= 350 + 14 (add the products)
= 364

Therefore, the value of 7 × 52 is 364.

Example 2: Find the value of -41 × 102

We can use the distributive property to simplify this expression as follows:

-41 × 102 = -41 × (100 + 2) (102 can be expressed as the sum of 100 and 2)
= -41 × 100 - 41 × 2 (multiply each addend separately)
= -4100 - 82 (add the products)
= -4182

Therefore, the value of -41 × 102 is -4182.

Properties of Multiplication of Integers

There are several properties of multiplication of integers that help us to manipulate and simplify expressions. The three most important properties are:

1. Closure Property: The product of any two integers is also an integer.

2. Commutative Property: The order of multiplication does not affect the product.

a × b = b × a

3. Associative Property: The grouping of factors does not affect the product.

(a × b) × c = a × (b × c)

These properties are useful when we need to simplify expressions involving multiplication of integers.

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