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.

\[ a \times b = b \times a \] Examples: 1. Real numbers: \[ 5 \times 7 = 7 \times 5 \] 2. Algebraic expressions: \[ (x^3 - 2) \times x = x \times (x^3 - 2) \]

3. Associative Property of Addition.

\[ (a + b) + c = a + (b + c) \] Examples:
1. real numbers \[ (2 + 3) + 6 = 2 + (3 + 6) \] 2. algebraic expressions \[ (x^3 + 2x) + x = x^3 + (2x + x) \]

4. Associative Property of Multiplication.

\[ (a \times b) \times c = a \times (b \times c) \] Examples:
1. Real numbers \[ (7 \times 3) \times 10 = 7 \times (3 \times 10) \] 2. Algebraic expressions \[ (x^2 \times 5x) \times x = x^2 \times (5x \times x) \]

5. Distributive Properties of Addition and Multiplication.

\[ a \times (b + c) = a \times b + a \times c \] and \[ (a + b) \times c = a \times c + b \times c \] Examples:
1. Real numbers \[ 2 \times (2 + 8) = 2 \times 2 + 2 \times 8 \] \[ (2 + 8) \times 10 = 2 \times 10 + 8 \times 10 \] 2. Algebraic expressions \[ x \times (x^4 + x) = x \times x^4 + x \times x \] \[ (x^4 + x) \times x^2 = x^4 \times x^2 + x \times x^2 \]

6. The reciprocal of a non zero

The reciprocal of a real number \(a\) is \[\frac{1}{a}\]. and \[ a \times \left(\frac{1}{a}\right) = 1\] Examples:
The reciprocal of \( 5 \) is \[ \frac{1}{5}\] and \[ 5 \times \left(\frac{1}{5}\right) = 1\].

7. The additive inverse of a number

The additive inverse of \(a \) is \[ - a \] and \[ a + (- a) = 0 \] Examples:
additive inverse of \( -6 \) is \[ -(-6) = 6 \] and \[ - 6 + (6) = 0 \]