Download Chapter 9 Roots and Radicals Section 9.3

Section 9.1
Finding Roots
OBJECTIVES
A
Find the square root of a
number.
B
Square a radical
expression.
OBJECTIVES
C
Classify the square root
of a number and
approximate it with a
calculator.
OBJECTIVES
D
E
Find higher roots of
numbers.
Solve an application
involving square roots.
DEFINITION
Square Root
If a is a positive real number,
a = b is the positive square root of
a so that b2 = a.
– a = b is the negative square root of
a so that b2 = a.
DEFINITION
Square Root
If a = 0, a = 0 and 0 = 0
since 02 = 0
RULE
Squaring a Square Root
When the square root of a nonnegative real number a is
squared, the result is that
positive real number:
2
2
 a  = a and  – a  = a.
RULE
Square Root of a
Negative Number
If a is negative, a is
not a real number.
Chapter 9
Roots and Radicals
Section 9.1
Exercise #1
Find.
a) 169
= 13
= 13
2
b)
–
=–
7
=–
9
49
81
72
92
Chapter 9
Roots and Radicals
Section 9.1
Exercise #2
Find the square of each radical expression.
b) x 2 + 7
a) – 121

= – 121
= 121

2


2
=  x +7


= x2 + 7
2
Chapter 9
Roots and Radicals
Section 9.1
Exercise #3
Classify each number as rational, irrational, or
not a real number, and simplify if possible.
a) 17
irrational
b) – 36
= – 6 rational
100
c) –100
Not real
d)
=
49
10
7
rational
Chapter 9
Roots and Radicals
Section 9.1
Exercise #4
Find each root, if possible.
a)
4
4
b) – 625
81
=
4 4
=3
3
4
=– 5
= –5
4
Chapter 9
Roots and Radicals
Section 9.1
Exercise #5
A diver jumps from a cliff 20 meters high. If the time t
(in seconds) it takes an object dropped from a distance
d (in meters) to reach the ground is given by:
t=
d
5
d = 20
How long does it take the
diver to reach the water?
t=
20
5
t= 4
The diver takes 2 seconds.
=2
Section 9.2
Multiplication and
Division of Radicals
OBJECTIVES
A
Multiply and simplify
radicals using the
product rule.
OBJECTIVES
B
Divide and simplify
radicals using the
quotient rule.
OBJECTIVES
C
Simplify radicals
involving variables.
OBJECTIVES
D
Simplify higher roots.
Product Rule for Radicals
If a and b are nonnegative
numbers,
a b= a
b
Quotient Rule for Radicals
If a and b are positive
numbers,
a =
b
a
b
Absolute Value for Radicals
For any real number a,
2
a = a
Properties of Radicals
For all real numbers where
the indicated roots exist,
na
b=
na
nb
a
and n =
b
na
nb
Chapter 9
Roots and Radicals
Section 9.2
Exercise #6
Simplify.
a)
125
b)
54
= 25 • 5
= 9•6
=5 5
=3 6
Chapter 9
Roots and Radicals
Section 9.2
Exercise #7
Multiply.
a)
3 • 11
b)
11 • y , y > 0
= 3 • 11
= 11 • y
=
=
33
11y
Chapter 9
Roots and Radicals
Section 9.2
Exercise #8
Simplify.
21 50
7
a)
16
7
=
16
=
7
4
b) 7 5
=
21
7
•
= 3 10
50
5
Chapter 9
Roots and Radicals
Section 9.2
Exercise #9
Simplify.
a) 144n 2 , n > 0 b) 32y 7 , y > 0
=
2 2
12 n
= 12n
=
42 • 2 • y 6 • y
= 4y
3
2y
Chapter 9
Roots and Radicals
Section 9.2
Exercise #10
Simplify.
a)
4
96
=
4
16 • 6
=
4
24 • 6
4
= 2 6
Section 9.3
Addition and
Subtractions of
Radicals
OBJECTIVES
A
Add and subtract like
radicals.
OBJECTIVES
B
Use the distributive
property to simplify
radicals.
OBJECTIVES
C
Rationalize the
denominator in an
expression.
PROCEDURE
Rationalizing Denominators
Method 1:
Multiply both numerator and
denominator of the fraction by the
square root in the denominator.
PROCEDURE
Rationalizing Denominators
Method 2:
Multiply numerator and
denominator by the square root of
a number that makes the
denominator the square root of a
perfect square.
Chapter 9
Roots and Radicals
Section 9.3
Exercise #11
Simplify.
a) 9 13 + 7 13
=
 9 + 7
= 16 13
13
b) 14 6 – 3 6
=  14 – 3  6
= 11 6
Chapter 9
Roots and Radicals
Section 9.3
Exercise #12
Simplify.
a)
28 + 63
= 4 7+ 9 7
= 2 7 +3 7
=  2 + 3 7
=5 7
Chapter 9
Roots and Radicals
Section 9.3
Exercise #13
Simplify.
a)
3


18 – 5 =
3 •
18 – 3 •
= 3•9•2 – 3 •5
=
9•3•2 –
= 3 6 – 15
3 5
5
Chapter 9
Roots and Radicals
Section 9.3
Exercise #14
Write
=
=
=
3
20
with a rationalized denominator.
3•5
20 • 5
15
100
15
10
Chapter 9
Roots and Radicals
Section 9.3
Exercise #15
Write
=
=
y
2
50
, y > 0 with a rationalized denominator.
y2 •
2
50 •
2
y 2
100
=
y 2
10
Section 9.4
Simplifying Radicals
OBJECTIVES
A
Simplify a radical
expression involving
products, quotients,
sums, or differences.
OBJECTIVES
B
Use the conjugate of a
number to rationalize the
denominator of an
expression.
OBJECTIVES
C
Reduce a fraction
involving a radical by
factoring.
RULES
Simplifying Radical Expressions
1. Whenever possible, write
the rational-number
representation of a radical
expression.
4
2
1
1
3
81 as 9, as , and as
9 3
8 2
RULES
Simplifying Radical Expressions
2. Use the product rule
x • y = xy to write
indicated products as a
single radical.
RULES
Simplifying Radical Expressions
6 instead of 2 • 3 and
2ab instead of 2a • b
Chapter 9
Roots and Radicals
Section 9.4
Exercise #16
Simplify.
a) 8 14 – 7 • 2
= 8 14 – 14
=  8 – 1 14
= 7 14
Simplify.
b)
12x 3
4x
, x >0
=
2
=
=
12x 3
4x 2
12
4
3x
• x3 – 2
Chapter 9
Roots and Radicals
Section 9.4
Exercise #17
Simplify.
3
a)
500
3
500
3
=
2
2
=
3
250
3
= 125 • 2
3
=5 2
Chapter 9
Roots and Radicals
Section 9.4
Exercise #18
Simplify.
b)

10 – 2 20
=

10
10 + 2 20
 –  2 20 
2
= 10 – 4  20 
= 10 – 80
= – 70

2

Chapter 9
Roots and Radicals
Section 9.4
Exercise #19
Simplify.
a)
11
=
3 +1
=
=
11 •



3 +1
11 •

11 3 – 11

3 –1

3 –1
3–1
2

3 –1
Chapter 9
Roots and Radicals
Section 9.4
Exercise #20
Simplify.
a)
– 6 + 18
3
=
=
=
9•2
–6 +
3
–6 +3 2

3
3 –2 +
= –2 +
3
2
2

Section 9.5
Applications
OBJECTIVES
A
Solve equations with
one square root term
containing the variable.
OBJECTIVES
B
Solve equations with two
square root terms
containing the variable.
OBJECTIVES
C
Solve an application.
PROCEDURE
Raising Both Sides of an
Equation to a Power
If both sides of the equation A = B
are squared, all solutions are
among the solutions of the new
2
2
equation A =B .
PROCEDURE
Solving Radical Equations
1. Isolate the square root term
containing the variable.
2. Square both sides of the
equation.
PROCEDURE
Solving Radical Equations
3. Simplify and repeat steps 1
and 2 if there is a square
root term containing the
variable.
PROCEDURE
Solving Radical Equations
4. Solve the resulting linear or
quadratic equation.
5. Check all proposed
solutions in the original
equation.
Chapter 9
Roots and Radicals
Section 9.5
Exercise #22
Solve.
x +4 –x =2
x +4 =x +2
x + 4 = x 2 + 4x + 4
0 = x 2 + 3x
0 = x  x + 3
x = 0 or x = – 3
Solve.
Check: x = 0, 0 + 4 – 0 ? 2
2 – 0 = 2

 
Check: x = – 3, – 3 + 4 – – 3 ? 2
1
+
3 ? 2
4  2
x = 0 or x = – 3
Chapter 9
Roots and Radicals
Section 9.5
Exercise #23
Solve.
y +3 =
2y + 1
y + 3 = 2y + 1
3 =y +1
2 =y y = 2
Check:
2 +3 ?
5 ?
2 •2+1
4+1
5 = 5 
Chapter 9
Roots and Radicals
Section 9.5
Exercise #24
Solve.
y + 6 – 3 2y – 5 = 0
y + 6 = 3 2y – 5

y + 6 = 9 2y – 5
y + 6 = 18y – 45
6 = 17y – 45
51 = 17y
3=y

Solve.
y + 6 – 3 2y – 5 = 0
Check:
3 +6 – 3 2 •3 – 5 ? 0
9 – 3 6–5 ? 0
3 – 3 1 ? 0
3 – 3? 0
3=y
0 = 0
