ZingPath: Exponents and Radicals

Exponentiation is the arithmetic operation used to describe successively multiplying the same quantity multiple times. The operation is usually denoted by superscripts, as in or Given a quantity a and positive integer n, a raised to the nth power is defined as follows:

In this case, a is called the base and n is called the exponent. In general, the exponent specifies the number of times that a occurs as a factor in the product. The exponent is sometimes referred to as the power. Several examples are computed below:

In the case that the exponent is one, the base occurs exactly once; therefore, for any quantity a. In the example above, we can use this reasoning to include .

Exponentiation inherits a number of properties from its multiplicative definition. These properties can help to simplify complicated expressions without resorting to the definition of an exponent. These properties can be justified in a straightforward fashion, by expanding the exponent into a product, regrouping the product according to the associative and distributive laws, and then recombining the expression into exponential expressions by using the definition of exponentiation.

For example, the product of powers law is a simple reformulation of the associative law of multiplication, which states that multiplication can be arbitrarily grouped, thereby allowing us to remove parentheses. In the following proof, this fact allows us to expand each exponent, remove the parentheses, count the factors of a, and then regroup into a simplified exponential expression.

This proof shows that the expression contains n + m factors equal to a because it is the product of n factors equal to a by m factors equal to a.

As a matter of well-justified mathematical convention, negative exponents have a special meaning. Specifically, if -n is a negative integer and a is nonzero, then describes the product containing n factors equal to

Several examples are computed below:

By this notation, the exponent -1 can be used to represent the multiplicative inverse of a nonzero number. Negative exponents follow the same properties as their nonnegative counterparts in addition to one extra property.

By using the negative exponents, along with the quotient of powers property, one can calculate the result of raising a nonzero base to the zeroth power by using the fact that 0 = 1 - 1.

This property is sometimes called the zero property of exponents.