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 + 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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