The continued product of a number multiplied with itself a number of times can be written as the number raised to the power a natural number, equal to the number of times the number is multiplied with itself.

To make calculations easier, we have few rules or laws of exponents:
•    Multiplying Powers with the same base
•    Dividing Powers with the same base
•    Power with Exponent zero
•    Power of a Power
•    Multiplying Powers with the same exponents
•    Dividing Powers with the same exponents

Let’s discuss each one of them in detail:

Law 1: Multiplying Powers with the same base

If ‘a’ is any non – zero rational number and m, n are natural numbers, then
a^m x aⁿ = a^{m + n}

Also, If ‘a’ is any non – zero rational number and m, n, p are natural numbers, then
a^m x aⁿ x a^p = a^{m + n + p}

Example: Simplify: 3² x 3⁵
= 3^{2 + 5}
= 3⁷

Law 2: Dividing Powers with the same base

If ‘a’ is any non – zero rational number and m, n are natural numbers such that m > n, then
                                                                               a^m ÷ aⁿ = a^{m – n} or a^m = a^{m – n}
                                                                                                          aⁿ

Example: Simplify: 9¹² ÷ 9¹⁰
= 9^{12 – 10}
= 9²

Law 3: Power with exponent zero

If ‘a’ is any non – zero rational number raise to power 0, then it is equal to 1
a⁰ = 1

Example: 7³ ÷ 7³
= 7^{3 – 3}
= 7⁰
= 1

Law 4: Power of a Power

If ‘a’ is any rational number different from zero and m, n are natural numbers, then
(a^m)ⁿ = a^{m x n} = (aⁿ)^m

Example: Simplify: (2³)⁴ = 2^{3 x 4} = 2¹²

Law 5: Multiplying Powers with the same exponents

If a, b are non – zero rational numbers and n is a natural number, then
aⁿ x bⁿ = (ab)ⁿ

Also, If a, b, c are non – zero rational numbers and n is a natural number, then
aⁿ x bⁿ x cⁿ = (abc)ⁿ

Example: 2⁵ x 3⁵
= (2 x 3)⁵
= 6⁵

Law 6: Dividing Powers with the same exponents

If a and b are non – zero rational numbers and n is a natural number, then
                                                                                              aⁿ = a ⁿ
                                                                               bⁿ = b

Example: (2/3)²
= (2 x 2)/(3 x3)
= 4/9
       

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Reference Links:

•     http://en.wikipedia.org/wiki/Exponentiation
•     http://en.wikipedia.org/wiki/Natural_number
•     http://en.wikipedia.org/wiki/Rational_number