# Basic Rules and Properties of Algebra

We list the basic rules and properties of algebra and give examples on how they may be used.

Let a, b and c be real numbers, variables or algebraic expressions.

## 1. Commutative Property of Addition.

a + b = b + a

Examples:

1. real numbers

2 + 3 = 3 + 2

2. algebraic expressions

x^{ 2} + x = x + x^{ 2}

## 2. Commutative Property of Multiplication.

a × b = b × a

Examples:

1. real numbers

5 × 7 = 7 × 5

2. algebraic expressions

(x^{ 3} - 2) × x = x × (x^{ 3} - 2)

## 3. Associative Property of Addition.

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

Examples:

1. real numbers

(2 + 3) + 6 = 2 + (3 + 6)

2. algebraic expressions

(x^{ 3} + 2 x) + x = x^{ 3} + (2 x + x)

## 4. Associative Property of Multiplication.

(a × b) × c = a × (b × c)

Examples:

1. real numbers

(7 × 3) × 10 = 7 × (3 × 10)

2. algebraic expressions

(x^{ 2} × 5 x) × x = x^{ 2} × (5 x × x)

## 5. Distributive Properties of Addition Over Multiplication.

a × (b + c) = a × b + a × c

and

(a + b) × c = a × c + b × c

Examples:

1. real numbers

2 × (2 + 8) = 2 × 2 + 2 × 8

(2 + 8) × 10 = 2 × 10 + 8 × 10

2. algebraic expressions

x × (x^{ 4} + x) = x × x^{ 4} + x × x

(x^{ 4} + x) × x^{ 2} = x^{ 4} × x^{ 2} + x × x^{ 2}

## 6. The reciprocal of a non zero real number a is **1/a**.

and
a × (1/a) = 1

Examples:

1. real numbers

reciprocal of 5 is 1/5 and 5 × (1/5) = 1

## 7. The additive inverse of a is **-a**.

a + (- a) = 0

Examples:

additive inverse of -6 is -(-6) = 6 and - 6 + (6) = 0

## 8. The additive identity is 0.

and
a + 0 = 0 + a = a

## 9. The multiplicative identity is 1.

and
a × 1 = 1 × a = a
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