Chapter 7
Radicals,
Radical Functions, and
Rational Exponents
7.1 Radical Expressions and
Functions

Square Root
If a >= 0,
then b >= 0, such that b2 = a, is the principal square
root of a
 √a=b


E.g.,
√25 = 5
 √100 = 10


4
2
2 2 4
---- = ----, because --- = ----49
7
7
49

9 + 16

= 25 = 5
9 + 16 = 3 + 4 = 7
Negative Square Root

25 = 5 ---- principal square root
- 25 = -5 ---- negative square root

Given: a



What is the square root of a?
Given: 25
What is the square root of 25?
 sqrt = 5, sqrt = -5, because 52 = 25, (-5)2 = 25

Square Root Function

f(x) = x
y
x
y= x
(x, y)
0
0
(0,0)
1
1
(1,1)
4
2
(4,2)
9
3
(9,3)
16
4
(16,4)
18
4.24
(18,4.2)
x
Excel Chart
Evaluating
a Square Root Function



Given: f(x) =
Find: f(3)
Solution:

12x – 20
f(3) = 12(3) – 20
= 36 – 20
=
16
=4
Domain of a Square Root Function



Given: f(x) = 3x + 12
Find the Domain of f(x):
Solution:

3x + 12 ≥ 0
3x ≥ -12
X ≥ -4
[-4, ∞)
Application

By 2005, an “hour-long” show on prime time TV was 45.4
min on the average, and the rest was commercials, plugs,
etc. But this amount of “clutter“ was leveling off in recent
years. The amount of non-program “clutter”, in minutes,
was given by:
M(x) = 0.7 x + 12.5

where x is the number of years after 1996.
What was the number of minutes of “clutter” in an hour
program in 2002?
Solution

Solution:
M(x) = 0.7 x + 12.5
 x = 2002 – 1996 = 6
M(6) = 0.7 6 + 12.5
~ 0.7(2.45) + 12.5
~ 14.2 (min)


In 2009?
x = 2009 – 1996 = 13
 M(13) = 0.7 13 + 12.5
~ 15 (min)

Cube Root and Cube Root Function

3


3


3

a = b,
means b3 = a
8 = 2,
because 23 = 8
-64
= -4
Because (-4)3 = -64
Cube Root Function

f(x) =
3
x
x
y =3 x
(x, y)
-27
-3
(-27,-3)
-8
-2
(-8,-2)
-1
-1
(-1,-1)
0
0
(0,0)
1
1
(1,1)
8
2
(8,2)
27
3
(27,3)
30
3.1
(30,3.1)
Simplifying Radical Expressions
3
-64x3 = 3 (-4x)3 = -4x
4

81 = 4 (3)4 = 3
 4 -81 = x has no solution in R,



since there is no x such that x4 = -81
In general
 n -a
 n -a
has an nth root when n is odd
has no nth root when n is even
7.2 Rational Exponents

What is the meaning of 71/3?
x = 71/3 means
 x3 = (71/3)3 = 7


Generally, a1/n is number such that

(a1/n)n = a
Your Turn
Simplify

1.
2.
3.
4.
641/2
(-125)1/3
(6x2y)1/3
(-8)1/3
Solutions

1.
2.
3.
4.
8
-5
3
6x2y
-2
Solve

10002/3



= (10001/3)2 = 102 = 100
163/2
 (161/2)3 = 43 = 64
-323/5
 -(321/5)3 = -(2)3 = -8
Your Turn

What is the difference
between -323/5 and (-32) 3/5
 between -163/4 and (-16) 3/4

Simplify

61/7 · 64/7


+ 4/7) =
32x1/2
--------16x3/4


= 6(1/4
= 2x(1/2 – 3/4) = 2x-1/4
(8.33/4)2/3

65/7
= 8.3(3/4 ∙ 2/3)
= 8.31/2
Simplify

49-1/2


(8/27)-1/3


= 1/(-64)2/3 = 1/((-64)1/3)2 = 1/(-4)2 = 1/16
(52/3)3


= 1/(8/27)1/3 = (27/8)1/3 = 271/3/81/3 = 3/2
(-64)-2/3


= (72)-1/2 = 7-1 = 1/7
= 52/3 ∙ 3 = 52 = 25
(2x1/2)5
 25x1/2 · 5 = 32x5/2
7.3 Multiplying & Simplifying
Radical Expressions

Product Rule
a · n b = n ab
or
 a1/n · b1/n = (ab)1/n
 Note: Factors have same order of root.
 n

E.g,


n
25
n
4
=
25 · 4 = 100 = 10
2000 = 400 · 5 = 400 · 5 = 20 5
Simplify Radicals by Factoring

√(80)


3
√(40)


= √(8 · 2 · 5) = √(23 · 2 · 5)
= √(24 · 5) = 4√(5)
3
3
= √(8 · 5) = √(23 · 5)
3
= 2√(5)
√(200x4y2)

= √(5 · 40x4y2) = √(5 · 5 · 8x4y2)
= √(52 · 22 · 2x4y2)
= 5 · 2x2y√(2) = 10x2y√(2)
Simplify Radicals by Factoring

5
√(64x3y7z29)
=
5
√(32 · 2x3y5y2z25z4)
5
= √(25y5z25 · 2x3y2z4)
5
5
= 2yz √(2x3y2z4)
Multiplying & Simplifying

√(15)·√(3)
=
√(45) = √(9·5) = 3√(5)
4
4
3
2
 √(8x y )·√(8x5y3)
4
=
√(64x8y5)
=
4
2
2x y√(4y)
4
= √(16·4x8y4y)
Application

Paleontologists use the function
W(x) = 4√(2x)
to estimate the walking speed of a dinosaur, W(x),
in feet per second, where x is the length, in feet, of
the dinosaur’s leg.
What is the walking speed of a dinosaur whose leg
length is 6 feet?

W(x) = 4√(2x)

W(6) =
=
=
=
~
~
4√(2·6)
4√(12)
4√(4·3)
8√(3)
8√(1.7)
14 (ft/sec)
(humans: 4.4 ft/sec walking
22 ft/sec running)
Your Turn

Simplify the radicals
 √(2x/3)·√(3/2)
 = √((2x/3)(3/2)) = √x
4
4
 √(x/3)·√(7/y)
4
4
 = √((x/3)(7/y)) = √(7x/3y)
3
 √(81x8y6)
3
3
6
2
6
2
2
 = √(27·3x x y )= 3x y √(3x2)
3
 √((x+y)4)
3
3
3
 =√((x+y) (x+y))= (x+y)√(x+y)
7.4 Adding, Subtracting, & Dividing
Adding (radicals with same indices & radicands)

8√(13) + 2√(13)


3
3
3
7√(7) – 6x√(7) + 12√(7)


= √(13) · (8 + 2) = 10√(13)
4
3
3
= √(7) ·(7 – 6x + 12) = (19 – 6x)√(7)
4
4
7√(3x) - 2√(3x) + 2x2√(3x)
4
4
2
2
 = √(3x) ·(7 – 2 + 2x ) = (5 + 2x ) √(3x)
Adding

7√(18) + 5√(8)


= 7√(9·2) + 5√(4·2) = 7·3 √(2) + 5·2√(2)
= 21√(2) + 10√(2) = 31√(2)
√(27x) - 8√(12x)

= √(9·3x) - 8√(4·3x) = 3√(3x) – 8·2√(3x)
= √(3x)·(3 – 16) = -13√(3x)
3
 √(xy2)

3
3
+ √(8x4y5)
√(xy2)
3
2
3
√(8x3y3xy2)
=
+
= √(xy ) (1 + 2xy)
3
= (1 + 2xy) √(xy2)
=
3
√(xy2)
3
+ 2xy √(xy2)
Dividing Radical Expressions


Recall: (a/b)1/n = (a)1/n/(b)1/n
(x2/25y6)1/2


=(x2)1/2 / (25y6)1/2
=x/5y3
(45xy)1/2/(2·51/2)
= (1/2) ·(45xy/5)1/2 = (1/2) ·(9·5xy/5)1/2
= (1/2) ·3(xy)1/2
= (3/2) ·(xy)1/2
(48x7y)1/3/(6xy-2)1/3
7
-2 1/3
 = ((48x y)/6xy ))
= (8x6y3)1/3


= 2x2y
7.5 Rationalizing Denominators


Given: 1
√(3)
Rationalize the denominator—get rid of the
radical in the denominator.
1 √(3) √(3)
=
√(3) √(3)
3
Denominator Containing 2 Terms


Given:
8
3√(2) + 4
Rationalize denominator


Recall: (A + B)(A – B) = A2 – B2
8
3√(2) – 4
8(3√(2) – 4)
=
3√(2) + 4 3√(2) – 4
(3√(2) )2 – (4)2
24 √(2) - 32
8(3 √(2) – 4)
=
18 – 16
12 √(2) - 16
=
2
Your Turn

Rationalize the denominator

2 + √(5)
√(6) - √(3)
 2+√(5)
√(6)+√(3)
2√(6)+2√(3)+√(5)√(6)+√(5)√(3)
=
√(6) - √(3) √(6)+√(3)
2√(6) + 2√(3) + √(30) +√(15)
=
3
6–3
7.6 Radical Equations


Application
A basketball player’s hang time is the time in the
air while shooting a basket. It is related to the
vertical height of the jump by the following
formula:
t = √(d) / 2
A Harlem Globetrotter slam-dunked while he
was in the air for 1.16 seconds. How high did
he jump?
Solving Radical Equations

√(x) = 10


(√(x))2 = 102
x = 100
√(2x + 3) = 5

(√(2x + 3) )2 = 52
(2x + 3) = 25
2x = 22
x = 11
Check
√(2x + 3) = 5
√(2(11) + 3) = 5
√(22 + 3) = 5
√(25)
=5
5
=5
?
?
?
yes
Solve

√(x - 3) + 6 = 5

√(x - 3) = -1
(√(x - 3))2 = (-1)2
(x – 3) = 1
x=4
Check:
√(x - 3) + 6 = 5
√(4 - 3) + 6 = 5 ?
√(1) + 6 = 5 ?
1+6
= 5 ? False
Thus, there is no solution to
this equation.
Your Turn

Solve: √(x – 1) + 7 = 2

√(x – 1) = -5
(√(x – 1))2 = (-5)2
x – 1 = 25
x = 26
Check:
√(x – 1) + 7
√(26 – 1) + 7
√(25) + 7
5+7
=2
=2 ?
=2 ?
= 2 ? False
Thus, there is no solution to
this equation.
Your Turn

Solve: x + √(26 – 11x) = 4

√(26 – 11x) = 4 – x
(√(26 – 11x))2 = (4 – x)2
26 – 11x = 16 – 8x + x2
0 = x2 + 3x – 10
x2 + 3x – 10 = 0
(x – 2)(x + 5) = 0
x–2=0
x=2
x+5=0
x = -5
Check -5:
√(26 – 11x) = 4 – x
√(26 – 11(-5)) = 4 – (-5) ?
√(26 + 55) = 4 + 5
?
√(81)
=9
?
9
= 9 True
Check 2:
√(26 – 11x) = 4 – x
√(26 – 11(2)) = 4 – 2 ?
√(4)
=2
?
2
=2
True
Solution: {-5, 2}
Hang Time in Basketball

A basketball player’s hang time is the time spent
in the air when shooting a basket. It is a
function of vertical height of jump.
√(d)
t = ----- where t is hang time in sec and
2 d is vertical distance in feet.

If Michael Wilson of Harlem Globetrotters had
a hang time of 1.16 sec, what was his vertical
jump?
Hang Time

√(d)
t = ----2
2t
= √(d)
2(1.16) = √(d)
2.32 = √(d)
(2.32)2 = (√(d))2
5.38 = d
7.7 Complex Numbers

What kind of number is x = √(-25)?


Imaginary Unit i


x2 = -25?
i = √(-1), i 2 = -1
Example
√(-25) = √((25)(-1)) = √(25)√(-1) = 5i
 √(-80) = √((80)(-1)) = √((16 · 5)(-1))
= 4√(5)i
= 4i √(5)

Your Turn

Express the following with i.
1.
2.
3.
4.
√(-49)
√(-21)
√(-125)
-√(-300)
Complex Numbers


Comlex number has a Real part and an
Imaginary part of the form: a + bi
Example
1. 2 + 3i
2. -4 + 5i
3. 5 – 2i
Adding and Subtracting Complex
Numbers


(5 – 11i) + (7 + 4i)
= 5 – 11i + 7 + 4i
= 12 – 7i
(2 + 6i) – (12 – 4i)
= 2 + 6i – 12 + 4i
= -10 + 10i
Multiplying Complex Numbers


4i(3 – 5i)
= 12i – 20i2
= 12i – 20(-1)
= 12 + 12i
(5 + 4i)(6 – 7i)
= 5·6 – 5 ·7i + 4i· 6 – 4 ·7i2
= 30 – 35i + 24i – 28(-1)
= 30 – 11i + 28
= 58 – 11i
Multiplying
1. √(-3) √(-5)
= i√(3) · i√(5)
= i2 √(15)
= -√(15)
2. √(-5) √(-10)
= i√(5) · i√(10)
= i2 √(50)
= -√(50)
= -√(25 · 2)
= -5√(2)
Conjugates and Division




Given: a + bi
Conjugate of a + bi: a – bi
Conjugate of a – bi: a + bi
Why conjugates?
(a + bi)(a – bi)
= (a)2 – (bi)2
= a2 – b2i2
= a2 + b2
(3 + 2i)(3 – 2i) = 9 – (2i)2= 9 – 4(-1) = 13
Multiplying a complex number by its conjugate results in
a real number.
Dividing Complex Numbers


Express 7 + 4i
-------as a + bi
2 – 5i
7 + 4i (7 + 4i) (2 + 5i)
14 + 35i + 8i + 20
-------- = ---------- · ----------- = -----------------------2 – 5 i (2 – 5i) (2 + 5i)
4 + 25
34 – 43i
= ------------29
Your Turn


6 + 2i
-------4 – 3i
6 + 2i (4 + 3i) 24 + 18i + 8i + 6i2
= ---------- · ---------- = ------------------------(4 – 3i) (4 + 3i)
16 + 9
(18 + 26i)
= ------------25
Your Turn


5i – 4
------3i
(5i – 4) -3i
-15i2 + 12i
= --------- · ----- = ------------------3i
-3i
-9i2
15 + 12i 3(5 + 4i) 5 + 4i
= ------------ = ----------- = --------9
9
3
Powers of i
 i2
i3
= -1
= (-1)i = -i
i4 = (-1)2 = 1
i5 = (i4)i = i
i6 = (-1)3 = -1
i7 = (i6)i = -i
i8 = (-1)4 = 1
i9 = (i8)i = i
i10 = (-1)5 = -1
Your Turn

Simplify
 i17

i17 = i16i = (i2)8i = i
 i50
 i50
= (i2)25 = (-1)25 = -1
 i35
 i35
= (i34)i = (i2)17i = (-1)17i = -i
Application



Electrical engineers use the Ohm’s law to relate
the current (I, in amperes), voltage (E, in volts),
and resistence (R, in ohms) in a circuit:
E = IR
Given: I = (4 – 5i) and R = (3 + 7i), what is E?
E = (4 – 5i)(3 + 7i) = 12 + 28i - 15i - 35i2
= 47 + 13i (volts)