We list the basic rules and propeties 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 mutliplicative identity is 1.
and
a * 1 = 1 * a = a
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