OBJECTIVE 4 Find nth roots of nth powers. Consider the expression.
At first glance, you may think that it is equivalent to a. However, this is not necessar-
ily true. For example, consider the following.
then
then Instead of , we get 6,
the absolute valueof -6.If a= - 6 , 2 a^2 = 21 - 622 = 236 = 6. -^6
If a= 6 , 2 a^2 = 262 = 236 = 6.
2 a^2
SECTION 8.1 Radical Expressions and Graphs 431
NOW TRY
EXERCISE 5
Simplify each root.
(a) (b)
(c) (d)
(e) 23 x^18 (f ) 24 t^20
- 241 - 1024 - 2 m^8
281 - 228 231 - 923
Since the symbol represents the nonnegativesquare root, we express with
absolute value bars, as |a|, because amay be a negative number.
2 a^22 a^2
For any real number a,
That is, the principal square root of a^2 is the absolute value of a.
2 a^2 a.
2 a^2If nis an evenpositive integer, then
If nis an oddpositive integer, then
That is, use the absolute value symbol when nis even. Absolute value is not
used when nis odd.
2 nana.
2 nana.
2 nanSimplifying Square Roots by Using Absolute ValueFind each square root.
(a) (b)
(c) 2 k^2 = |k| (d) 21 - k 22 =|-k|=|k| NOW TRY
272 =| 7 |= 7 21 - 722 =|- 7 |= 7
EXAMPLE 4
Simplifying Higher Roots by Using Absolute ValueSimplify each root.
(a) nis even. Use absolute value.
(b) nis odd.
(c) nis even. Use absolute value.
(d) For all
No absolute value bars are needed here, because is nonnegative for any real
number value of m.
(e) because
(f )
We use absolute value to guarantee that the result is not negative (because is
negative when xis negative). If desired |x^3 |can be written as x^2 #|x|.
x^3
24 x^12 = |x^3 |
23 a^12 =a^4 , a^12 = 1 a^423.
m^2
- 2 m^4 =-|m^2 |=-m^2 m, |m^2 |=m^2.
- 241 - 924 = -|- 9 |=- 9
251 - 425 = - 4
261 - 326 =|- 3 | = 3
EXAMPLE 5
NOW TRY
EXERCISE 4
Find each square root.
(a) (b)
(c) 2 z^2 (d) 21 - z 22
2112 21 - 1122
NOW TRY ANSWERS
- (a) 11 (b) 11 (c)
(d) - (a) 2 (b) (c)
(d)-m^4 (e)x^6 (f )|t^5 |- 9 - 10
|z||z|NOW TRY