Overview
As division is the inverse of multiplication, so division of exponents is the inverse of multiplication of exponents. There are properties for the division of exponents similar to the properties for the multiplication of exponents.
Quotient of Powers
Recall that finding a product of powers with the same base is expressed by the rule in algebra language of xmxn = xm +n. This concept was developed by using the definition of exponents as repeated multiplication. If x3 is x·x·x and x2 is x·x, then x3x2 is (x·x·x)(x·x). What about (x3)/(x2)? That also can be simplified by using the definition of exponents, as (x·x·x)/(x·x) is x3-2 or x. In the language of algebra, xm/xn equals xm-n, when x is a real number not equal to zero and m and n are integers.
Positive Power of a Quotient
Recall also that any expression (ab)n can be expanded to anbn. For example, an expression such as (3x)3 means 3·3·3·x·x·x or 27x3. Similarly, a quotient such as (2/5)4 means (2/5)(2/5)(2/5)(2/5) or (24)/(54). The quotient is in simplest form with the exponents, or it can be multiplied out as 16/625. In the language of algebra, (a/b)n equals an/bn, if a and b are real numbers not equal to zero, and n is a positive integer.
Negative Power of a Quotient
Remember that x-n is the reciprocal of xn or 1/xn. For example, 5-3 is the same thing as 1/53 or 1/125. Similarly, the reciprocal of 5/6 is 1/(5/6) or 6/5. Suppose that the value of x is the fraction a/b. In that case, x-n would equal 1/(a/b)n, or 1/(an/bn), or bn/an, which is the reciprocal of an/bn. In the language of algebra, if a and b are real numbers not equal to zero and n is a positive integer, then (a/b)-n is equal to (b/a)n, which is equal to bn/an.
Applications of Exponents
Exponents are often used in scientific notation to express numbers that are very large or very small. For example, the distance from the Sun to the Earth is about 9.3 x 107 miles. However, the size of an average dust particle is about 7.53 x 10-10 kg. Exponents are also used in other applications, including economics, measurement, engineering, and accounting.
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