### 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:

**associative property of addition**states that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:

**identity property of addition**states that when adding zero to any any given number, the sum is the given number itself:

**additive identity**is the number zero, which has the property that $a+0=a$ for any real number $a$. This is stated as the

**identity property of addition**, which is the sum of zero and any number is the given number:

**additive inverse**is the number $-a$. It is also called the opposite of $a$. The sum of a number and its additive inverse is the additive identity, zero. This value is called the inverse, or opposite, because it reverses the addition. For any number, when adding $a$ and then adding $-a$, the result is the original number:

### Addition Properties

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. |

$a+b=b+a$ |
$a+(b+c)=(a+b)+c$ |
$a+0=a$ |

$9+5=5+9$ |
$5+(7+6)=(5+7)+6$ |
$26+0=26$ |

### Additive Inverses

Value | Additive Inverse | Number Plus Inverse |
---|---|---|

$a$ |
$-a$ |
$a+(-a)=0$ |

$97$ |
$-97$ |
$97+(-97)=0$ |

$-\frac{4}{5}$ |
$-\left(-\frac{4}{5}\right)=\frac{4}{5}$ |
$-\frac{4}{5}+\frac{4}{5}=0$ |

$0$ |
$-0=0$ |
$0+0=0$ |

### 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:

**associative property of multiplication**states that when multiplying three or more numbers, the product does not change based on the way the numbers are grouped:

**multiplicative identity**is the number 1, which has the property that $a\cdot1=a$ for any real number $a$. This is stated as the

**identity property of multiplication**, which is the product of 1 and any number is the given number:

**multiplicative inverse**is the number $\frac{1}{a}$. It is also called the reciprocal of $a$. The product of a number and its multiplicative inverse is the multiplicative identity, 1. This value is called the inverse because it reverses the multiplication. For any number, when multiplying by $a$ and then multiplying by $\frac{1}{a}$, the result is the original number:

### Multiplication Properties

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. |

$ab=ba$ |
$a(bc)=(ab)c$ |
$a\cdot1=a$ |

$4\cdot12=12\cdot4$ |
$3(2\cdot5)=(3\cdot2)5$ |
$7\cdot1=7$ |

### Multiplicative Inverses

Value | Multiplicative Inverse | Number Times Inverse |
---|---|---|

$a$ |
$\frac{1}{a}$ |
$a\cdot\frac{1}{a}$ |

$8$ |
$\frac{1}{8}$ |
$8\cdot\frac{1}{8}=1$ |

$-3$ |
$-\frac{1}{3}$ |
$-3\cdot\left(-\frac{1}{3}\right)=1$ |

$\frac{2}{3}$ |
$\frac{1}{\left(\frac{2}{3}\right)}=\frac{3}{2}$ |
$\frac{2}{3}\cdot\frac{3}{2}=1$ |

$0$ |
$\frac{1}{0}$ is undefined. So, there is no multiplicative inverse of zero. | There is no number that can be multiplied by zero to produce 1. |

### Distributive Property

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.

### Distributive Property of Multiplication over Addition

Add, then multiply. | Multiply, then add. |
---|---|

$\begin{gathered}2(4+6)\\2(10)\\20\end{gathered}$ |
$\begin{gathered}2(4+6)\\2(4)+2(6)\\8+12\\20\end{gathered}$ |

- The term $ac$ is the product of the First terms of $({\color{#c42126}a}+b)({\color{#c42126}c}+d)$.
- The term $ad$ is the product of the Outside terms of $({\color{#c42126}a}+b)(c+{\color{#c42126}d})$.
- The term $bc$ is the product of the Inside terms of $(a+{\color{#c42126}b})({\color{#c42126}c}+d)$.

The term $bd$ is the product of the Last terms of $(a+{\color{#c42126}b})(c+{\color{#c42126}d})$.

### 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.