C h a p t e r dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557 ’ 9.1 9.2 9.3 9.4 9.5 9.6 Radicals Rational Exponents Adding, Subtracting, and Multiplying Radicals Quotients, Powers, and Rationalizing Denominators Solving Equations with Radicals and Exponents Complex Numbers 9 Radicals and Rational Exponents Just how cold is it in Fargo, North Dakota, in winter? According to local meteorol­ ogists, the mercury hit a low of –33°F on January 18, 1994. But air temperature alone is not always a reliable indicator of how cold you feel. On the same date, the average wind velocity was 13.8 miles per hour. This dramatically affected how cold people felt when they stepped outside. High winds along with cold temper­ atures make exposed skin feel colder because the wind significantly speeds up the loss of body heat. Meteorologists use the terms “wind chill factor,”“wind chill index,” and “wind chill temperature” to take into account both air temperature and wind velocity. Through experimentation in Ant­ arctica, Paul A. Siple developed a formula in the 1940s that measures the wind chill from the velocity of the wind and the air temperature. His complex formula involving the square root of the velocity of the wind is still used today to calculate wind chill temper­ atures. Siple’s formula is unlike most scientific formulas in that it is not based on theory. Siple experimented with various formulas involving wind velocity and temperature until he found a formula that seemed to predict how cold the air felt. W in d ch ill te m pe ra tu re ( F ) fo r 25 F a ir te m pe ra tu re 25 20 15 10 5 0 5 10 15 Wind velocity (mph) 5 10 15 20 25 30 Siple s formula is stated and used in Exercises 111 and 112 of Section 9.1. dug84356_ch09a.qxd 9/14/10 2:11 PM Page 558 → → → 558 Chapter 9 Radicals and Rational Exponents 9-2 9.1 Radicals In Section 4.1, you learned the basic facts about powers. In this section, you will study roots and see how powers and roots are related. In This Section U1V Roots U2V Roots and Variables U3V Product Rule for Radicals U4V Quotient Rule for Radicals U1V Roots U5V Domain of a Radical We use the idea of roots to reverse powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or Function -3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2 and -2 are fourth roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one real cube root of 8 and only one real cube root of -8. The cube root of 8 is 2 and the cube root of -8 is -2. nth Roots If a = bn for a positive integer n, then b is an nth root of a. If a = b2, then b is a square root of a. If a = b3, then b is the cube root of a. If n is a positive even integer and a is positive, then there are two real nth roots of a. We call these roots even roots. The positive even root of a positive number is called the prin ...

1. C
h
a
p
t
e
r
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557
’
9.1
9.2
9.3
9.4
9.5
9.6
Radicals
Rational Exponents
Adding, Subtracting, and

2. Multiplying Radicals
Quotients, Powers,
and Rationalizing
Denominators
Solving Equations with
Radicals and Exponents
Complex Numbers
9
Radicals and Rational
Exponents
Just how cold is it in Fargo, North Dakota, in winter? According
to local meteorol-
ogists, the mercury hit a low of –33°F on January 18, 1994. But
air temperature
alone is not always a reliable indicator of how cold you feel. On
the same date,
the average wind velocity was 13.8 miles per hour. This
dramatically affected how
cold people felt when they stepped outside. High winds along
with cold temper-
atures make exposed skin feel colder because the wind
significantly speeds up
the loss of body heat. Meteorologists use the terms “wind chill
factor,”“wind chill

3. index,” and “wind chill temperature” to take into account both
air temperature
and wind velocity.
Through experimentation in Ant-
arctica, Paul A. Siple developed a
formula in the 1940s that measures the
wind chill from the velocity of the wind
and the air temperature. His complex
formula involving the square root of
the velocity of the wind is still used
today to calculate wind chill temper-
atures. Siple’s formula is unlike most
scientific formulas in that it is not
based on theory. Siple experimented
with various formulas involving wind
velocity and temperature until he
found a formula that seemed to predict
how cold the air felt.
W

4. in
d
ch
ill
te
m
pe
ra
tu
re
(
F
)
fo
r
25
F
a
ir
te
m
pe
ra
tu

5. re
25
20
15
10
5
0
5
10
15
Wind velocity (mph)
5 10 15 20 25 30
Siple s formula is stated
and used in Exercises 111
and 112 of Section 9.1.
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 558
→
→

6. →
558 Chapter 9 Radicals and Rational Exponents 9-2
9.1 Radicals
In Section 4.1, you learned the basic facts about powers. In this
section, you will
study roots and see how powers and roots are related.
In This Section
U1V Roots
U2V Roots and Variables
U3V Product Rule for Radicals
U4V Quotient Rule for Radicals U1V Roots
U5V Domain of a Radical We use the idea of roots to reverse
powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or
Function
-3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2
and -2 are fourth
roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one
real cube root of 8 and
only one real cube root of -8. The cube root of 8 is 2 and the
cube root of -8 is -2.
nth Roots
If a = bn for a positive integer n, then b is an nth root of a. If a
= b2, then b is a
square root of a. If a = b3, then b is the cube root of a.

7. If n is a positive even integer and a is positive, then there are
two real nth roots of
a. We call these roots even roots. The positive even root of a
positive number is called
the principal root. The principal square root of 9 is 3 and the
principal fourth root of
16 is 2, and these roots are even roots.
If n is a positive odd integer and a is any real number, there is
only one real nth
root of a. We call that root an odd root. Because 25 = 32, the
fifth root of 32 is 2 and
2 is an odd root.
We use the radical symbol Va to signify roots.
U Helpful Hint V V n aa
The parts of a radical:
If n is a positive even integer and a is positive, then V n a a
denotes the principal nth Radical
root of a.Index
V n
symbol
aa If n is a positive odd integer, then V n a a denotes the nth
root of a.
Radicand If n is any positive integer, then V
n a 0 = 0.
We read V n a a as “the nth root of a.” In the notation V n a a, n
is the index of the
radical and a is the radicand. For square roots the index is

8. omitted, and we simply
write Va a.
E X A M P L E 1 Evaluating radical expressions
Find the following roots:
a) Va25
3
-b) Va27
c) V6 64a
d) -Va 4
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 559
9-3 9.1 Radicals 559
Solution
a) Because 52 = 25, V25a = 5.
b) Because (-3)3 = -27, V3 -27a = -3.
c) Because 26 = 64, V6 64a = 2.
d) Because V4a = 2, -V4a = -(V4a) = -2.

9. U Calculator Close-Up V
We can use the radical symbol to find
a square root on a graphing calcula-
tor, but for other roots we use the xth
root symbol as shown. The xth root
symbol is in the MATH menu.
Now do Exercises 1–16
CAUTION In radical notation, V4a represents the principal
square root of 4, so
V4a = 2. Note that -2 is also a square root of 4, but Va4 * -2.
Note that even roots of negative numbers are omitted from the
definition of nth
roots because even powers of real numbers are never negative.
So no real number can
be an even root of a negative number. Expressions such as
V-a9, V-81 and V-64
4 a, 6 a
are not real numbers. Square roots of negative numbers will be

10. discussed in Section 9.6
when we discuss the imaginary numbers.
U2V Roots and Variables
A whole number is a perfect square if it is the square of another
whole number. So 9
is a perfect square because 32 = 9. Likewise, an exponential
expression is a perfect
square if it is the square of another exponential expression. So
x10 is a perfect square
because (x5)2 = x10. The exponent in a perfect square must be
divisible by 2. An expo-
nential expression is a perfect cube if it is the cube of another
exponential expression.
So x21 is a perfect cube because (x7)3 = x21. The exponent in a
perfect cube must be
divisible by 3. The exponent in a perfect fourth power is
divisible by 4, and so on.

11. 2 4 6 8 10 12
Perfect squares x , x , x , x , x , x , . . . Exponent divisible by 2
3 6 9 12 15 18
Perfect cubes x , x , x , x , x , x , . . . Exponent divisible by 3
4 8 12 16 20 24
Perfect fourth powers x , x , x , x , x , x , . . . Exponent
divisible by 4
To find the square root of a perfect square, divide the exponent
by 2. If x is non-
negative, we have
2 4 2 6 3
Vx Vx , a = x and so on.a = x, a = x Vx ,
We specified that x was nonnegative because Vx a = x3 are not
correcta2 = x and Vx6
if x is negative. If x is negative, x and x3 are negative but the
radical symbol with an
even root must be a positive number. Using absolute value
symbols we can say that
2 6

12. Vx a = lx3l for any real numbers.a = lxl and Vx
To find the cube root of a perfect cube, divide the exponent by
3. If x is any real
number, we have
3 3 3
3 6 2 9 3
Vx Vx , Vx , and so on.a = x, a = x a = x
Note that both sides of each of these equations have the same
sign whether x is posi-
tive or negative. For cube roots and other odd roots, we will not
need absolute value
symbols to make statements that are true for any real numbers.
We need absolute value
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 560
560 Chapter 9 Radicals and Rational Exponents 9-4
symbols only when the result of an even root has an odd
exponent. For example,
Vm30 = lm6 a 5l for any real number m.

13. E X A M P L E 2 Roots of exponential expressions
Find each root. Assume that the variables can represent any real
numbers. Use absolute
value symbols when necessary.
a) Va2a b) Vx22a c) V4 w40a d) V3 t18a e) V5 s30a