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
- (a) 10 (b) 5 (c)^14 (d)0.3