Properties of Addition
Three important properties of addition can be used to simplify expressions.The commutative property of addition states that when adding real numbers, the sum does not change based on the order of the numbers:
|Commutative Property of Addition||Associative Property of Addition||Identity Property of Addition|
|Order does not affect the sum.||Grouping does not affect the sum.||Adding zero does not affect the sum.|
|Value||Additive Inverse||Number Plus Inverse|
Properties of Multiplication
The properties of multiplication are similar to the properties of addition and are also used to simplify expressions.The commutative property of multiplication states that when multiplying real numbers, the product does not change based on the order of the numbers:
|Commutative Property of Multiplication||Associative Property of Multiplication||Identity Property of Multiplication|
|Order does not affect the product.||Grouping does not affect the product.||Multiplying by 1 does not affect the product.|
|Value||Multiplicative Inverse||Number Times Inverse|
|is undefined. So, there is no multiplicative inverse of zero.||There is no number that can be multiplied by zero to produce 1.|
The distributive property states that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products.The value that is multiplied can be on the left side of the expression, such as:
Distributive Property of Multiplication over Addition
|Add, then multiply.||Multiply, then add.|
- The term is the product of the First terms of .
- The term is the product of the Outside terms of .
- The term is the product of the Inside terms of .
The term is the product of the Last terms of .
Applying Properties of Operations
If an expression has more than one operation, performing the operations in a different order can produce different results. So, there is an agreed-upon order to ensure the result is always the same. The order of operations is a set of rules indicating which calculations to perform first to simplify a mathematical expression.
1. Simplify expressions within grouping symbols, such as parentheses , brackets , or braces , or expressions in the numerator or denominator of a fraction or under a radical. For nested grouping symbols , start with the innermost set, and work to the outside.
2. Simplify any exponents or radicals in the expression.
3. Multiply and divide, working from left to right across the expression.
4. Add and subtract, working from left to right across the expression.
Properties of operations can be used with the order of operations to simplify expressions.