Intermediate Algebra (11th edition)

(Marvins-Underground-K-12) #1

428 CHAPTER 8 Roots, Radicals, and Root Functions


OBJECTIVES OBJECTIVE 1 Find roots of numbers. Recall that We say that 6


squaredis 36. The opposite (or inverse) of squaringa number is taking its square root.


236 = 6 , because 62 = 36


62 =36.


Radical Expressions and Graphs


8.1


1 Find roots of
numbers.
2 Find principal roots.
3 Graph functions
defined by radical
expressions.
4 Find nth roots of
nth powers.
5 Use a calculator to
find roots.

It is customary
to write
rather than 22.

2

We extend this discussion to cube roots 23 ,fourth roots 24 , and higher roots.


The nth root of a, written , is a number whose nth power equals a. That is,


n
2 ab means bna.

n

2 a


2 na

The number ais the radicand,nis the indexor order,and the expression is a


radical.


Index Radical
symbol

Radicand
Radical

2


n

a


2


n

a


NOW TRY
EXERCISE 1
Simplify.


(a) (b)


(c) (d) 23 0.027
B


4


1


256


231000 24625


⎧⎪⎨⎪⎩

Simplifying Higher Roots

Simplify.


(a) , because. (b) because


(c) , because (d) , because


(e) because


(f ) 24 0.0016= 0.2,because 1 0.2 24 = 0.0016. NOW TRY


a


2


3


b


3

=


8


27


.


B


3


8


27


=


2


3


,


2416 = 2 24 = 16. 2532 = 2 25 = 32.


2364 = 4 43 = 64 23125 =5, 53 = 125.


EXAMPLE 1


OBJECTIVE 2 Find principal roots.If nis even, positive numbers have two nth


roots. For example, both 4 and are square roots of 16, and 2 and are fourth


roots of 16. In such cases, the notation represents the positive root, called the


principal root,and - 2 represents the negative root.


n

a


2


n

a


- 4 - 2


nth Root

Case 1 If nis evenand ais positive or 0,then


represents the principal nth rootof a,


and represents the negative nth rootof a.


Case 2 If nis evenand ais negative,then


is not a real number.


Case 3 If nis odd,then


there is exactly one real nth root of a, written 2.


n

a


2


n

a





n

2 a


n

2 a


NOW TRY ANSWERS



  1. (a) 10 (b) 5 (c)^14 (d)0.3

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