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Exponents and Radicals

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



The number b is called an nth 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 nth root of a number is the nth 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 4th 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.

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