Exponents

       In an earlier lesson, we have learned that for any nonzero rational number a and a natural number n, the product

a × a × a × ... × a,

that is, the continued product of a multiplied by itself n times, is written as aⁿ. It is known as the nth power of a and is read as "a raised to the power n." The rational number a is called the base and n is called the exponent or index. This notation of writing the product of a rational number by itself several times is called exponential notation or power notation.

       For example, 3/4 × 3/4 × 3/4 × 3/4= (3/4)⁴. The rational number 3/4 is the base and 4 is exponent. Consider (3/5)⁴=3/5 × 3/5 × 3/5 × 3/5=(3 × 3 × 3 × 3) ÷ (5 × 5 × 5 × 5)=3⁴/5⁴.

Properties of Exponents

If a/b and c/d are any two rational numbers and m is a positive integer
(a/b × c/d)^m=(a/b)^m × (c/d)^m.


If a/b is any rational number, and m and n are positive integers
(a/b)^m × (a/b)ⁿ=(a/b)^{m + n}.


If a/b is any rational number, and m and n are any teo positive integer
(a/b)^m ÷ (a/b)ⁿ=(a/b)^{m - n}.



Zero Exponent

If a/b is any nonzero rational number

(a/b)⁰=1


For any nonzero rational number a and a positive integer n
a^-n=1/aⁿ.