# Basic Rules and Properties of Algebra

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

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

## 1. Commutative Property of Addition.

a + b = b + aExamples:

1. real numbers

2 + 3 = 3 + 2

2. algebraic expressions

x

^{ 2}+ x = x + x

^{ 2}

## 2. Commutative Property of Multiplication.

a × b = b × aExamples:

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 × cand

(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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