# 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.

•

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