There are many times in algebra when you need to simplify an expression. The properties of real numbers provide tools to help you take a complicated expression and simplify it.
The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to work.
Step-by-step explanation:
Commutative Property of Multiplication
For any real numbers a and b, a · b = b · aOrder does not matter as long as the two quantities are being multiplied together. This property works for real numbers and for variables that represent real numbers.
just as subtraction is not commutative, neither is division commutative. 4 ÷ 2 does not have the same quotient as 2 ÷ 4.Associative Property of Addition
For any real numbers a, b, and c, (a + b) + c = a + (b + c).
Associative Property of Multiplication
For any real numbers a, b, and c, (a • b) • c = a • (b • c).Multiplication has an associative property that works exactly the same as the one for addition. The associative property of multiplication states that numbers in a multiplication expression can be regrouped using parentheses. For example, the expression below can be rewritten in two different ways using the associative property.
The parentheses do not affect the product, the product is the same regardless of where the parentheses are.
The distributive property of multiplication is a very useful property that lets you rewrite expressions in which you are multiplying a number by a sum or difference. The property states that the product of a sum or difference, such as 6(5 – 2), is equal to the sum or difference of products, in this case, 6(5) – 6(2).
6(5 – 2) = 6(3) = 18
65) – 6(2) = 30 – 12 = 18
The distributive property of multiplication can be used when you multiply a number by a sum. For example, suppose you want to multiply 3 by the sum of 10 + 2.3(10 + 2) = ?
According to this property, you can add the numbers 10 and 2 first and then multiply by 3, as shown here: 3(10 + 2) = 3(12) = 36. Alternatively, you can first multiply each addend by the 3 (this is called distributing the 3), and then you can add the products. This process is shown here.
3 (10 + 2) = 3(12) = 36
3(10) + 3(2) = 30 + 6 = 36
The products are the same.
since multiplication is commutative, you can use the distributive property regardless of the order of the factors.
