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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 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 1given 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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