Exponents and Radicals

Radicals and Properties of Radicals
Radicals (or roots) are, in effect, the opposite of exponents. In other words, the n^th root of a number a is a
number b such that



The number b is called an n^th root of a. The number n is referred to as the index of the radical (if no index
appears, n is understood to be 2). The principal n^th root of a number is the n^th root of a which has the same
sign as a . For example both 2 and - 2 satisfy , but 2 is the (principal) square root of 4.
Examples:
• since
• since (Note also, but 2 is the principal 4^th root
• since
• is not a real number and we will say that it does not exist. (In this course we won’t learn how to
take an eventh power of a negative number .)

Radicals are used to define rational exponents :



The notation is extremely useful, and we encourage you to use it whenever you have to simplify
expressions involving radicals.
Examples:

Since radicals are nothing more than rational exponents , many of the properties of exponents also apply to
radicals.

Property	Example
5a If n is odd 5b If n is even

The following list is a restatement of these properties, but in exponential notation . You need to be familiar
with both radical and exponential notation, and be able to convert between the two.

Property	Example
5a If n is odd 5b If n is even

Examples:
• (refer to Property 5b)
• (refer to property 1-given the right hand side)
• (refer to property 1)
 There is no answer as we cannot take the square root of -16.
•