Download Lecture note 8

8
Roots, Radicals, and Root
Functions
8.1 Radical Expressions
and Graphs
R.1 Fractions
Objectives
1.
2.
3.
4.
5.
Find roots of numbers.
Find principal roots.
Graph functions defined by radical expressions.
Find nth roots of nth powers.
Use a calculator to find roots.
Section18.1,
1
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1-1
Find Roots of Numbers
Section28.1,
2
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1-2
Find Roots of Numbers
Example 1
Previously, we found square roots of positive numbers
such as
Simplifying Higher Roots
and
3
64 = 4, because 43 = 64.
4
81 = 3, because 34 = 81.
5
100,000 = 10, because 105 = 100,000.
because 10 • 10 = 100 and 11 • 11 = 121.
The number a is the
radicand, n is the index,
or order, and the
expression n a is a
radical.
Section38.1,
3
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1-3
Section48.1,
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Finding Roots
Example 2
Find each root.
Finding Principal Roots
25
25
4
3
Section58.1,
5
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1-5
Because the radicand, 25, is positive, there
are two square roots, –5 and 5. The
principal root is 5.
5
5
The index is even and the radicand is
negative, so this is not a real number.
81
64
Here, we want the negative square root, –5.
4
The index is odd, so the root is –4 since
(–4)3 = –64.
Section68.1,
6
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1
Graphs of Functions Defined by Radical Expressions
Graphs of Functions Defined by Radical Expressions
Example 3a
Graph f x
x 2 by creating a table of values. Give
the domain and range.
The domain of f x
principal square root.
x is [0,∞) because x is the
Section78.1,
7
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1-7
For the radicand to be nonnegative, we must have x – 2 ≥ 0.
Therefore, the domain is [2, ∞). The range is [0.∞).
Section88.1,
8
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1-8
Graphs of Functions Defined by Radical Expressions
Simplifying Square Roots Using Absolute Value
Example 3b
3
Graph f x
x by creating a table of values. Give the
domain and range.
3
x is any real number (–∞, ∞). The
The domain of f x
range is any real number (–∞, ∞).
Section98.1,
9
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1-9
Simplifying Square Roots Using Absolute Value
Section
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1-10
Simplifying Higher Roots Using Absolute Value
Example 4
Find each square root. Note that m is a real number.
11
112
11
2
11
11
m
m2
m
11
2
m
m
Section
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104
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Section
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1-12
2
Simplifying Higher Roots Using Absolute Value
Use a Calculator to Find Roots
Example 5
Numbers such 25 and 3 27 are rational, but radicals are
often irrational numbers. We can use a calculator to find the
approximate roots such as 17, 25, and 4 8.
Find each square root.
8
8
5
7
7
5
m6
4
m8
5
5
Thus,
5
m6
m2
m2
Section
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1-13
Section
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1-14
Using a Calculator to Find the Velocity of an Orbiting Body
8.2 Rational Exponents
R.1 Fractions
Example 7
Objectives
When calculating the velocity of a body in elliptical orbit at a
distance r from the focus, in terms of the semimajor axis a,
we encounter the expression
2 1
.
r a
1. Use exponential notation for nth roots.
2. Define and use expressions of the form am/n.
3. Convert between radicals and rational
exponents.
4. Use the rules for exponents with rational
exponents.
Evaluate the expression when r = 10,260 and a = 14,460.
2
r
1
a
2
10,260
1
14,460
Substitute
Use a calculator
0.011215
Section
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Exponents of the Form a1/n
EXAMPLE 1
Example 1
If
Evaluating Exponentials of the Form a1/n
Evaluate each expression.
a1/n
n
Section
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1-16
a
is a real number, then
a1/n =
n
a
.
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1-17
(a)
271/3 =
(b)
3
27
= 3
641/2 =
64
= 8
(c)
–6251/4 =
–
(d)
(–625)1/4
=
4
625 = –5
4
–625
is not a real number because the radicand,
–625, is negative and the index is even.
Section
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1-18
3
Evaluating Exponentials of the Form a1/n
Caution on Roots
Continued.
CAUTION
Evaluate each expression.
Notice the difference between parts (c) and (d) in Example 1. The radical
in part (c) is the negative fourth root of a positive number, while the radical
in part (d) is the principal fourth root of a negative number, which is
not a real number.
(e)
(f)
(–243)1/5 =
4
25
5
–243
1/2
4
25
=
= –3
2
5
=
1
Continued.
(c)
–6251/4 =
(d)
(–625)1/4
–
=
4
625 = –5
4
–625
is not a real number because the radicand,
–625, is negative and the index is even.
Section
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Exponents of the Form am/n
Section
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EXAMPLE
Example 22
Evaluating Exponentials of the Form am/n
Evaluate each exponential.
am/n
If m and n are positive integers with m/n in lowest
terms, then
am/n = ( a1/n ) m,
a1/n
(a)
253/2 = ( 251/2 )3 = 53 = 125
(b)
322/5 = ( 321/5 )2 = 22 = 4
(c)
–274/3 = –( 27)4/3 = –( 271/3 )4 = –(3)4 = –81
(d)
(–64)2/3 = [(–64)1/3 ]2 = (–4)2 = 16
(e)
(–16)3/2 is not a real number, since (–16)1/2 is not a real number.
a1/n
provided that
is a real number. If
is not a real
number, then am/n is not a real number.
Section
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Example 3
Evaluating Exponentials with Negative
Rational Exponents
Evaluate each exponential.
(a)
Section
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Evaluating Exponentials with Negative
Rational Exponents
Continued.
Evaluate each exponential.
32–4/5
(b)
8
27
–4/3
=
By the definition of a negative exponent,
32–4/5 =
Since 324/5 =
5
4
32
= 24
1
324/5
8
27
1
324/5
4/3
=
3
We could also use the rule
.
= 16,
= 32–4/5 =
1
8
27
=
–4/3
=
27
8
4/3
=
b
a
1
8
27
–m
3
4
a
b
=
27
8
=
m
4
=
1
2
3
4
=
1
16
81
=
81
16
here, as follows.
3
2
4
=
81
16
1
.
16
Section
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Section
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4
Alternative Definition of am/n
Caution on Roots
am/n
If all indicated roots are real numbers, then
am/n = ( a1/n ) m = ( a m ) 1/n.
Section
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Radical Form of am/n
Section
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Converting between Rational Exponents
and Radicals
Example 4
Write each exponential as a radical. Assume that all variables represent
positive real numbers. Use the definition that takes the root first.
Radical Form of am/n
If all indicated roots are real numbers, then
am/n =
n
n
am = (
a )
m
= ( 6 10 )5
(a)
151/2
=
(c)
4n2/3
= 4( 3 n )2
(d)
7h3/4 – (2h)2/5 = 7( 4 h )3 – ( 5 2h )2
(e)
g–4/5
15
(b)
105/6
.
In words, raise a to the mth power and then take the nth
root, or take the nth root of a and then raise to the mth
power.
=
1
g4/5
=
1
( 5 g )4
Section
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Converting between Rational Exponents
and Radicals
Continued.
In (f) – (h), write each radical as an exponential. Simplify.
Assume that all variables represent positive real numbers.
(f)
Section
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1-28
33
=
331/2
Rules for Rational Exponents
Rules for Rational Exponents
Let r and s be rational numbers. For all real numbers a
and b for which the indicated expressions exist:
(g)
3
76
=
76/3
(h)
5 m5
=
m, since m is positive.
=
72
=
ar · as = ar + s
49
( ar ) s = ar s
Section
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1-29
a–r =
1
ar
( ab ) r = ar br
ar
= ar – s
as
a
b
r
=
ar
br
a
b
–r
r
= br
a
a–r =
1
a
r
.
Section
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1-30
5
Applying Rules for Rational Exponents
Example 5
Continued.
Write with only positive exponents. Assume that all
variables represent positive real numbers.
63/4 · 61/2 = 63/4 + 1/2
(a)
(b)
= 65/4
32/3 = 32/3 – 5/6 = 3–1/6 = 1
31/6
35/6
Product rule
Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all
variables represent positive real numbers.
m1/4 n–6
m–8 n2/3
(c)
–3/4
Quotient rule
=
(m1/4)–3/4 (n–6)–3/4
( m–8)–3/4 (n2/3)–3/4
=
m–3/16 n9/2
m6 n–1/2
=
m–3/16 – 6 n9/2 – (–1/2)
=
m–99/16 n5
=
n5
m99/16
Section
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Continued.
Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all
variables represent positive real numbers.
(d)
x3/5(x–1/2 – x3/4) = x3/5 · x–1/2 – x3/5 · x3/4
= x3/5 + (–1/2) – x3/5 + 3/4
Distributive property
Power rule
Quotient rule
Definition of negative
exponent
Section
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Caution on Converting Expressions to Radical Form
CAUTION
Use the rules of exponents in problems like those in Example 5. Do not
convert the expressions to radical form.
Product rule
= x1/10 – x27/20
Do not make the common mistake of multiplying exponents in the
first step.
Section
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1-33
EXAMPLE
Example 66
Applying Rules for Rational Exponents
Rewrite all radicals as exponentials, and then apply the rules for rational
exponents. Leave answers in exponential form. Assume that all variables
represent positive real numbers.
(a)
4
a3 ·
3
a2 = a3/4 · a2/3
= a3/4 + 2/3
=
a9/12 + 8/12
=
a17/12
Section
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1-34
Continued.
Rewrite all radicals as exponentials, and then apply the rules for rational
exponents. Leave answers in exponential form. Assume that all variables
represent positive real numbers.
4
Convert to rational exponents.
(b)
c
=
c1/4
c3/2
Convert to rational exponents.
=
c1/4 – 3/2
Quotient rule
=
c1/4 – 6/4
Write exponents with a common
denominator
=
c–5/4
=
1
c5/4
c3
Product rule
Write exponents with a common
denominator
Section
8.1,of
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Applying Rules for Rational Exponents
Definition of negative exponent
Section
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6
8.3 Simplifying Radical
Expressions
R.1 Fractions
EXAMPLE
Continued.6
Applying Rules for Rational Exponents
Rewrite all radicals as exponentials, and then apply the rules for rational
exponents. Leave answers in exponential form. Assume that all variables
represent positive real numbers.
5
(c)
3 x2
5
=
x2/3
( x2/3 )1/5
=
x2/15
=
Objectives
1.
2.
3.
4.
Use the product rule for radicals.
Use the quotient rule for radicals.
Simplify radicals.
Simplify products and quotients of radicals with
different indexes.
5. Use the Pythagorean theorem.
6. Use the distance formula.
Section
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1-37
Section
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1-38
Use the Product Rule for Radicals
Use the Product Rule for Radicals
Example 1
Multiply. Assume that all variables represent positive real
numbers.
6
5
13ab
6 5
2c
30
26abc
Section
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1-39
Use the Product Rule for Radicals
Section
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1-40
Use the Quotient Rule for Radicals
Example 2
Multiply. Assume that all variables represent positive real
numbers.
3
6
4
3
15 x 3
11
6
3
4 11
5 xy 2
6
3
44
4
75x y 2
Cannot be simplified using the product
rule because the indexes, are different.
Section
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1-41
Section
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7
Use the Quotient Rule for Radicals
Example 3
Simplify. Assume that all variables represent positive real
numbers.
81
16
5
11
7
11
49
27
8
3
9
4
81
16
11
49
3
27
8
3
y 10
243
Simplifying Radicals
27
3
5
y 10
5
243
3
2
8
3
2
y2
3
Section
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1-43
Section
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Simplifying Radicals
Simplifying Radicals with Variables
Example 4
Be careful to leave the
5 inside the radical.
Simplify.
28
6125
4 7
4
2 7
7
53 7 2
125 49
Example 5 Simplify. Assume all variables represent
positive real numbers.
52 5 72
5 7 5
3
2 62 2 x 2 x
2 72x 3
2 62
35 5
x2
2 6 x
2x
2x
54x 3
3
3
3
3
3
Cannot be simplified further.
3
2x 3
3
3
3
2
3
x3
2 x
12x 2x
3x 3 2
3
128
3
64 2
3
64
3
2
43 2
Section
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1-45
Section
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Simplifying Radicals – Smaller / Different Indices
Simplifying Radicals – Smaller / Different Indices
Example 6
Simplify. Assume that all variables represent real numbers.
12
Section
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1-47
74
7 4/12
71/3
3
7
Section
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1-48
8
Simplifying Radicals – Smaller / Different Indices
Pythagorean Theorem
Example 7
5
Simplify
5
5
The Pythagorean theorem relates lengths of the sides
of a right triangle.
3.
5
51/2
55/10
3
31/5
2/10
3
10
10
10
55
3
2
10
3125
9
Section
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49
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1-49
Pythagorean Theorem
Example 8
Use the Pythagorean theorem to find the length of side a.
Section
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1-50
The Distance Formula
9
c2
a2
b2
a
56
92
a2
52
a
4 14
a
2
a2
81 25
a
56
a
4
a
90º
5
14
2 14
Section
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1-51
Section
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1-52
8.4 Adding and Subtracting Radical
R.1 Fractions
Expressions
Objectives
The Distance Formula
Example 9
Find the distance between (1, 6) and (4, –2).
1. Simplify radical expressions involving addition
and subtraction.
Let (x1, y1) = (1,6) and (x2, y2) = (4, –2)
d
x2
x1
4 1
3
2
2
y2
2
2 6
8
y1
2
2
2
9 64
73
Section
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1-53
Section
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9
Simplify Radical Expressions Involving Addition and Subtraction
Adding and Subtracting Radicals
Example 1
Add or subtract to simplify each radical expression.
6 5 6
3 5
1 5 2
2 6
2 11 5 11
5
8 6
6
3 5
5
27
2 5
3
128
32 3
32
11
64 2
4 3
4 2
4 5 3 11
Section
8.1,of
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55
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1-55
Adding and Subtracting Radicals
Add or subtract to simplify each radical expression.
x 2y 3
Section
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1-56
Adding and Subtracting Radicals with Greater Indexes
Example 2
Continued
3 8x 2y
Add or subtract. Assume all variables represent positive
numbers.
x 50y
10 3 16a 4 b 2
3 4 2 x2 y
3 2 x 2y
6 x 2y
6x
x 2 y2 y
x y
xy 2y
xy 5 x
x
xy
2y
11a 3 54ab 2
x 25 2 y
x 5
10 3 8 2 a3 a b 2
3
10 2 a 2ab
2y
20a 3 2ab 2
20a 33a
5 x 2y
2y or 2y x
Adding and Subtracting Radicals with Fractions
5
18
50
3
3
36 2
25
6 2
5
18 2
5
5
5
3
9 2
25 2
11 a 3
3
2a b 2
33a 3 2ab 2
3
2ab 2
Section
8.1,of
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1-58
Adding and Subtracting Radicals with Fractions
Continued.
Add or subtract. Assume all variables represent positive
numbers.
3
3
7
11
12 3 9 7 3 12 12 7 7 11
Quotient rule
9
y
y
3
3
y
y 12
12 3 7
y3
3 2
5 2
18 2
5
11a 3 27 2a b 2
xy
Example 3
Add or subtract. Assume all variables represent positive
numbers.
72
25
2
53a 3 2ab 2
2y
Section
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1-57
3
16 2
8 2 4 2
3 3 3
2 11 5 11
3 1 5
These cannot be combined.
3
15
5
18 2 15
5
Section
8.1,of
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1-59
12y 3 7
y4
7 3 11
y4
7 3 11
y4
12y 3 7 7 3 11
y4
Section
8.1,of
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10
8.5 Multiplying and Dividing Radical
R.1 Fractions
Expressions
Objectives
Multiply Radical Expressions
1. Multiply radical expressions.
2. Rationalize denominators with one radical term.
3. Rationalize denominators with binomials
involving radicals.
4. Write radical quotients in lowest terms.
Multiplication of expressions that contain radicals is very
similar to multiplication of polynomials.
7
5
35
7 5
3 2
3 5 2 7
5
3 2
3 2 6
3 2 6 5 12
7
5 7
3 5 12
6 35
2 18 5 36
2 9 2 5 36
6 2 30
Section
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1-61
Section
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62
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1-62
Multiply Radical Expressions
Multiply Radical Expressions
Example 1
Multiplication of expressions that contain radicals is very
similar to multiplication of polynomials.
Multiply by using the FOIL method.
2
x
3
x
3
x
x
x
3 x
x 6 x
2
3 x
2 y
2
3 x
3
3
x
x 3
32
3 3
FOIL Method
9
2
2 3 x 2 y
9 x 12 xy
2 y
2
x
4y
2
x 6 x
9
Section
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1-63
Rationalize Denominators with One Radical Term
A simplified radical expression has no radical in the
denominator. The process of removing a radical from the
denominator of a fractional expression is called rationalizing
the denominator.
Section
8.1,of
Slide
64
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1-64
Rationalize Denominators with One Radical Term
Example 2
Rationalize each denominator.
5
6
5
6
6
5 6
6
6
2 3
2 3
2
5 2
5 2
2
2 6
5 2
2 6
10
7
12
7
7
3
12
2 3
3
21
2 3
Section
8.1,of
Slide
65
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1-65
6
5
21
6
Section
8.1,of
Slide
66
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1-66
11
Rationalize Denominators with Square Roots
Example 3
Simplify.
98c 3
,d
d7
98c
d7
0
3
2 49 c
2
Rationalize Denominators with Cube Roots
Example 4
To rationalize a denominator with a cube root, we must
multiply the numerator and denominator by a number that
will result in a perfect cube.
c
d2 d2 d2 d
7c 2c
d
d3 d
d
7
7
3
4
3
2
7
2
3
2
2
3
2
73 2
3
2
3
73 2
2
23
7c 2cd
d3 d
7c 2cd
d4
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1-67
Rationalize Denominators with Binomials Involving Radicals
Section
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1-68
Rationalize Denominators with Binomials Involving Radicals
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1-69
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Rationalize Denominators with Binomials Involving Radicals
Rationalize Denominators with Binomials Involving Radicals
Example 5
Continued.
Rationalize the denominator.
Rationalize the denominator.
1
1
5
2
5
2
5
2
5
2
5
2
25 5 2 5 2 2
5
2
25 2
7
6
7
2
6
2
6
2
6
2
7
6
2
6 2
7
6
2
4
5
2
23
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Section
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12
Writing Radical Quotients in Lowest Terms
Writing Radical Quotients in Lowest Terms
Example 6
Continued.
Write the quotient in lowest terms.
2 3
2 3
2 3
2
2
2 3
2
2
2 3
2 3
4 3 4 6 2
4 3 2
14 4 6
10
2
2
12 4 6
10
2 7 2 6
2 5
CAUTION
Be careful to factor
before writing a
quotient in lowest
terms.
Write the quotient in lowest terms.
2
8
2
8
2
8
2
8
2
8
2
8
2 2
2 8
16 8
2 8
2 2 2 8 4 8
6
6 2 2
6 2 12
6
6
2
7 2 6
5
2 2
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2 2
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8.6 Solving Equations
with Radicals
R.1 Fractions
Power Rule for Solving Equations with Radicals
Objectives
Power Rule for Solving Equations with Radicals
1. Solve radical equations using the power rule.
2. Solve radical equations that require additional
steps.
3. Solve radical equations with indexes greater
than 2.
4. Use the power rule to solve a formula for a
specified variable.
If both sides of an equation are raised to the same power, all solutions of
the original equation are also solutions of the new equation.
Read the power rule carefully; it does not say that all solutions
of the new equation are solutions of the original equation. They
may or may not be. Solutions that do not satisfy the original
equation are called extraneous solutions; they must be
discarded.
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Example 1
Caution on Checking Solutions
Solve
CAUTION
When the power rule is used to solve an equation, every solution of the
new equation must be checked in the original equation.
Using the Power Rule
5x – 6 = 7.
Use the power rule and square both sides to get
5x – 6
2
= 72
5x – 6 = 49
5x = 55
x = 11
Add 6.
Divide by 5.
To check, substitute the potential solution in the original equation.
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1-77
Section
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13
Continued.
Solve
Using the Power Rule
Solving an Equation with Radicals
Solving an Equation with Radicals
5x – 6 = 7.
Check:
5x – 6
5 · 11 – 6
Step 1
Isolate the radical. Make sure that one radical term is alone on
one side of the equation.
Step 2
Apply the power rule. Raise both sides of the equation to a
power that is the same as the index of the radical.
Step 3
Solve. Solve the resulting equation; if it still contains a radical,
repeat Steps 1 and 2.
Step 4
Check all potential solutions in the original equation.
= 7
= 7 ? Let x = 11.
7 = 7 True
Since 11 satisfies the original equation, the solution set is { 11 }.
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1-79
Section
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1-80
Using the Power Rule
Example 2
Caution on Incorrect Solution Sets
Solve
CAUTION
Remember to check Step 4 or you may get an incorrect solution set.
2x – 9 + 1 = 0.
Step 1
To isolate the radical on one side, subtract 1 from each side.
Step 2
Now square both sides.
2x – 9
2x – 9
Step 3
= –1
2
2x – 9
Solve.
= (–1) 2
= 1
2x = 10
x = 5
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1-81
Continued.
Solve
Step 4
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Using the Power Rule
Note on Identifying No Solution
2x – 9 + 1 = 0.
NOTE
Check the potential solution, 5, by substituting it in the original
equation.
2x – 9 + 1 = 0
We could have determined after Step 1 that the equation in Example 2
has no solution because the expression on the left cannot be negative.
2 · 5 – 9 + 1 = 0 ? Let x = 5.
1 + 1 = 0
Solve
False
This false result shows that 5 is not a solution of the original equation;
it is extraneous. The solution set is ∅.
Step 1
2x – 9 + 1 = 0.
To isolate the radical on one side, subtract 1 from each side.
2x – 9
= –1
The expression on the left cannot be negative.
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1-83
Section
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14
Example 3
Finding the Square of a Binomial
Solve
Finding the Square of a Binomial
Recall that
(x +
y)2
=
x2
+ 2xy +
y2.
Using the Power Rule (Squaring a Binomial)
9 – x = x + 3.
Step 1
The radical is alone on the left side of the equation.
Step 2
Now square both sides.
9 – x
9 – x
Step 3
2
= (x + 3)2
= x2 + 6x + 9
The new equation is quadratic, so get 0 on one side.
Twice the product of 3 and x.
0 = x2 + 7x
Subtract 9 and add x.
0 = x(x + 7)
Factor.
x = 0 or x + 7 = 0
Zero-factor property
x = –7
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1-85
Section
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1-86
Using the Power Rule (Squaring a Binomial)
Continued.
Caution on the Middle Term
CAUTION
Step 4
Check each potential solution, 0 and –7, in the original equation.
If x = –7, then
If x = 0, then
9 – x
= x + 3
9 – x
9 – 0
= 0 + 3 ?
9 – (–7) = (–7) + 3 ?
9
= 3
?
3
= 3
True
When a radical equation requires squaring a binomial as in Example 3,
remember to include the middle term.
(x + 3)2 = x2 + 6x + 9
= x + 3
16
= –4
?
4
= –4
False
The solution set is { 0 }. The other potential solution, –7, is extraneous.
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Example 4
Solve
Using the Power Rule (Squaring a Binomial)
x2 – 8x + 3 = x – 5.
Section
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1-88
Continued.
Solve
Square both sides.
Using the Power Rule; Squaring a Binomial
x2 – 8x + 3 = x – 5.
Check:
x2 – 8x + 3
x2 – 8x + 3
2
x2 – 8x + 3 = x – 5
= (x – 5) 2
= x2 – 10x + 25
112 – 8 · 11 + 3 = 11 – 5
Twice the product of 5 and x.
2x = 22
6 = 6
?
Let x = 11.
True
Subtract x2 and 3; add 10x.
The solution set of the original equation is { 11 }.
x = 11
Divide by 2.
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1-89
Section
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15
Example 5
Solve
Using the Power Rule (Squaring Twice)
Using the Power Rule (Squaring Twice)
Continued.
–x + 6 = 5.
x + 7 +
Start by isolating one radical on one side of the equation by subtracting
–x + 6 from each side. Then square both sides.
x + 7 +
This equation still contains a radical, so square both sides again. Before doing
this, isolate the radical term on the right.
x + 7
=
25 – 10
x + 7
=
31 – x – 10
2x – 24
=
–10
x – 12
=
–5
–x + 6
(x – 12)2 =
–5
–x + 6
–x + 6 = 5
x + 7 = 5 –
2
x + 7
x + 7
–x + 6
=
5 –
–x + 6
=
25 – 10
2
–x + 6 + (–x + 6)
–x + 6 + (–x + 6)
–x + 6
–x + 6
Subtract 31 and add x.
Divide by 2.
2
Square both sides again.
–x + 6.
Twice the product of 5 and
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1-91
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1-92
Using the Power Rule (Squaring Twice)
Using the Power Rule (Squaring Twice)
Continued.
Continued.
This equation still contains a radical, so square both sides again.
(x – 12)2 =
–5
–x + 6
(–5)2
2
Now finish solving the equation.
Square both sides again.
2
x2 – 24x + 144
=
x2 – 24x + 144
=
25 ( –x + 6 )
x2 – 24x + 144
=
–25x + 150
Distributive property
x2 + x – 6 =
0
Standard form
(x + 3)(x – 2) =
0
Factor.
–x + 6
(ab)2 = a2 b2
(x + 3)(x – 2) =
0
x + 3 = 0
or
x – 2 = 0
x = –3
or
x = 2
Zero-factor property
Section
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1-93
EXAMPLE 6 Using the Power Rule for a Power Greater
Example 6
than 2
Using the Power Rule; Squaring Twice
Continued.
Solve 3 3x – 2
Check each potential solution, –3 and 2, in the original equation.
If x = –3, then
x + 7 +
Section
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1-94
3 3x – 2 3 =
If x = 2, then
–x + 6
3 2x + 6.
=
Raise both sides to the third power.
= 5
x + 7 +
–x + 6
–(–3) + 6 = 5
2+ 7 +
–(2) + 6 = 5
3 2x + 6 3
3x – 2 = 2x + 6
= 5
x = 8
–3 + 7 +
4 +
9 = 5
5 = 5
9 +
Check this result in the original equation.
4 = 5
5 = 5
The solution set is { –3, 2 }.
3 3x – 2
=
3 2x + 6
3 3·8 – 2
=
3 2·8 + 6
?
3 22
=
3 22
True
Let x = 8.
The solution set is { 8 }.
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1-95
Section
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1-96
16
8.7 Complex Numbers
R.1 Fractions
Simplify Numbers of the Form
Objectives
1.
2.
3.
4.
5.
6.
, Where b >0
Simplify numbers of the form
b ,where b > 0.
Recognize subsets of the complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Simplify powers of i.
Section
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104
1-97
Simplify Numbers of the Form
, where b >0
Section
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1-98
Example 1 Simplify Numbers of the Form
where b >0
,
Write each number as a product of a real number and i.
3
i 3
4
i 4
9
8
2i
Section
8.1,of
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99
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99
104
1-99
i 9
i 8
3i
2i 2
Section
8.1, Slide
100
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100
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1-100
Multiplying Square Roots of Negative Numbers
Multiplying Square Roots of Negative Numbers
Example 2 Multiply.
7
11
i 7
11
Product rule for radicals
i 7 11
i 77
2
32
i 2 i 32
i 2 2 32 Product rule for radicals
1
64
8
Section
8.1, Slide
101
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101
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1-101
Section
8.1, Slide
102
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102
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1-102
17
Dividing Square Roots of Negative Numbers
162
i 162
2
147
i 147
3
i 3
2
i
Recognize Subsets of the Complex Numbers
Divide.
Example 3
162
147
2
162
i
2
3
147
3
i 81
49
9i
7
By combining the imaginary unit i and the real numbers,
a new set of numbers can be formed. This new set,
called the complex numbers, includes the real numbers
as a subset.
Section
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103
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103
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1-103
Section
8.1, Slide
104
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104
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1-104
Recognize Subsets of the Complex Numbers
Imaginary
part of the
number
Real part of
the number
Examples:
Adding and Subtracting Complex Numbers
Pure
imaginary
number
All written in standard form
Section
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105
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105
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1-105
Section
8.1, Slide
106
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106
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1-106
Adding and Subtracting Complex Numbers
Example 4
Example 5
Add.
5 3i
Adding and Subtracting Complex Numbers
6 9i
5 6
Subtract.
3 9 i
8 2i
11 6i
3 4i
8 3
5
8
7 5i
8 7
5i
2
4 i
2 4 i
5 2i
15 5i
Section
8.1, Slide
107
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107
of 104
1-107
Section
8.1, Slide
108
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108
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1-108
18
Multiplying Complex Numbers
Multiplying Complex Numbers
Example 6
Multiply.
5
6 3i
5
8i 3 2i
6 5 3i
30 15i
8i 3
8i 2i
24i 16i 2
24i 16
1
16 24i
Section
8.1, Slide
109
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109
of 104
1-109
Section
8.1, Slide
110
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110
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1-110
Multiplying Complex Numbers
Multiplying Complex Numbers
Continued.
Multiply.
1 2i 3 4i
13
1 4i
3 4i
6i
2i 3
8i
3 10i
8i 2
3 10i
8
2i
4i
2
1
3 10i 8
5 10i
Section
8.1, Slide
111
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111
of 104
1-111
Section
8.1, Slide
112
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112
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1-112
Dividing Complex Numbers
Dividing Complex Numbers
Example 7
Divide.
2 3i
1 i
2 3i
1 i
1 i
1 i
2 2i 3i 3i 2
1 i i i2
2 i
1
5 i
2
Section
8.1, Slide
113
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113
of 104
1-113
3
1
1
5
2
i
2
Section
8.1, Slide
114
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114
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1-114
19
Dividing Complex Numbers
Dividing Complex Numbers
Continued
Continued
Divide.
Divide.
2 i
3 2i
2 i
3 2i
3 2i
3 2i
7 4i
i
6 4i 3i
9 6i 6i
2i 2
4i 2
7 4i
i
6 7i
1
7i
4i 2
i2
7i
4
9 4
4 7i
13
2
1
4
13
7i
13
i
i
1
4 7i
1
Section
8.1, Slide
115
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115
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1-115
Section
8.1, Slide
116
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116
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1-116
Simplifying Powers of i
Simplifying Powers of i
Example 8
Find each power of i.
i 17
i i 16
i4
i 100
i
1
43
i
43
i4
i
25
1
25
1
i 3 i 40
4
i
i 1
1
1
1
i
3
i
4
10
i3 1
10
1
i3
1 i
i i
i
i2
Section
8.1, Slide
117
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117
of 104
1-117
i
1
i
Section
8.1, Slide
118
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118
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1-118
20