Download 7-1 Simplifying Radicals simplifying_radical_expressions

Chapter R: Reference: Basic Algebraic
Concepts
R.1
Review of Exponents and Polynomials
R.2
Review of Factoring
R.3
Review of Rational Expressions
R.4
Review of Negative and Rational Exponents
R.5
Review of Radicals
Copyright © 2007 Pearson Education, Inc.
Slide R-2
R.5 Review of Radicals
Radical Notation for a1/n
If a is a real number, n is a positive integer, and a1/n is
a real number, then
n
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a a
1/ n
.
Slide R-3
R.5
Review of Radicals
In the expression
•
n
n
a
is called a radical sign,
•
a is called the radicand,
•
n is called the index.
Copyright © 2007 Pearson Education, Inc.
Slide R-4
R.5 Evaluating Roots
Example Evaluate each root.
(a)
4
16
(b)
4
16
(c)
5
32
Solution
(a)
4
16  161/ 4  (24 )1/ 4  2
(b)
4
16 is not a real number.
(c)
5
32  [(2)5 ]1/ 5  2
Copyright © 2007 Pearson Education, Inc.
Slide R-5
R.5 Review of Radicals
Radical Notation for am/n
If a is a real number, m is an integer, n is a positive
integer, and n a is a real number, then
a
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m/ n

 
n
a
m
 n am .
Slide R-6
R.5 Converting from Rational Exponents
to Radicals
Example Write in radical form and simplify.
(b) (32) 4 / 5
(a) 8 2 / 3
(c) 3x 2 / 3
Solution
 8
(b) (32)  
2/3
(a) 8

2
3
4/ 5
(c) 3x
2/ 3
5
3 x
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3
2 4
2
32

4
 (2)4  16
2
Slide R-7
R.5 Converting from Radicals to Rational
Exponents
Example Write in exponential form.
(a)
4
x
( x  0)
5
(b) 10
 z
5
2
(c) 5 3 (2 x 4 ) 7
Solution
(a)
4
x x
5
5/ 4
( x  0)
(b) 10
 z
5
2
 10 z
2/ 5
(c) 5 3 (2 x 4 )7  5(2 x 4 )7 / 3  5  27 / 3 x 28/ 3
Copyright © 2007 Pearson Education, Inc.
Slide R-8
R.5 Review of Radicals
Evaluating
n
an
n
If n is an even positive integer, then
If n is an odd positive integer, then
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n
an  a .
an  a .
Slide R-9
R.5 Using Absolute Value to Simplify
Roots
Example Simplify each expression.
(a)
4
p4
(b)
16m8 r 6
(c)
6
(2) 6
Solution
(a)
4
16m8 r 6  (4m4 r 3 )2  4m4 r 3  4m4 r 3
(b)
(c)
p4  p
6
(2)  2  2
6
Copyright © 2007 Pearson Education, Inc.
Slide R-10
R.5 Review of Radicals
Rules for Radicals
For all real numbers a and b, and positive integers m and
n for which the indicated roots are real numbers,
n
a  n b  n ab
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n
a na
 n (b  0)
b
b
m n
a  mn a .
Slide R-11
R.5 Using the Rules for Radicals to
Simplify Radical Expressions
Example Simplify each expression.
6  54
(a)
(b)
3
m m
3
2
(c)
7
64
Solution
6  54  6  54  324  18
(a)
(b)
(c)
3
m  3 m2  3 m3  m
7
7
7


64
8
64
Copyright © 2007 Pearson Education, Inc.
Slide R-12
R.5 Simplifying Radicals
Simplified Radicals
An expression with radicals is simplified when the following
conditions are satisfied.
1. The radicand has no factor raised to a power greater than
or equal to the index.
2. The radicand has no fractions.
3. No denominator contains a radical.
4. Exponents in the radicand and the index of the radical
have no common factor.
5. All indicated operations have been performed (if possible).
Copyright © 2007 Pearson Education, Inc.
Slide R-13
R.5 Simplifying Radicals
Example Simplify each radical.
175
(a)
(b)
3
81x 5 y 7 z 6
Solution
175  25  7  25  7  5 7
(a)
(b)
3
81x y z  27  3  x  x  y  y  z
5
7 6
3
3
2
6
6
 3 27 x3 y 6 z 6 (3x 2 y )
 3xy 2 z 2 3 3x 2 y
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Slide R-14
R.5 Simplifying Radicals by Writing
Them with Rational Exponents
Example Simplify each radical.
(a)
6
2
(b)
3
6
x12 y 3 ( y  0)
Solution
(a)
6
3 3
(b)
6
x12 y 3  ( x12 y 3 )1/ 6  x 2 y 3/ 6  x 2 y1/ 2  x 2 y ( y  0)
2
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2/ 6
3
1/3
33
Slide R-15
R.5 Adding and Subtracting Like
Radicals
Example Add or subtract, as indicated. Assume all
variables represent positive real numbers.
(a) 7 2  8 18  4 72
(b)
98 x3 y  3x 32 xy
Solution
(a) 7 2  8 18  4 72  7 2  8 9  2  4 36  2
 7 2  83 2  46 2
 7 2  24 2  24 2
7 2
Copyright © 2007 Pearson Education, Inc.
Slide R-16
R.5 Adding and Subtracting Like
Radicals
Solution (b)
98 x3 y  3x 32 xy  49  2  x 2  x  y  3x 16  2  x  y
 7 x 2 xy  3x(4) 2 xy
 7 x 2 xy  12 x 2 xy
 19 x 2 xy
Copyright © 2007 Pearson Education, Inc.
Slide R-17
R.5 Multiplying Radical Expressions
Example Find each product.
(a)

2 3

8 5

(b)

7  10

7  10

Solution (a) Using FOIL,

2 3


8 5  2
 8 
2(5)  3 8  3(5)


 16  5 2  3 2 2  15
 4  5 2  6 2  15
 11  2
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Slide R-18
R.5 Multiplying Radical Expressions
Solution (b)

7  10

  7    10 
7  10 
2
2
 7  10
 3
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Slide R-19
R.5
Rationalizing Denominators
•
The process of simplifying a radical
expression so that no denominator contains a
radical is called rationalizing the
denominator.
•
Rationalizing the denominator is
accomplished by multiplying by a suitable
form of 1.
Copyright © 2007 Pearson Education, Inc.
Slide R-20
R.5
Rationalizing Denominators
Example Rationalize each denominator.
4
(a)
3
Solution
(a)
(b)
2
(
x

0)
3
x
(b)
4
4
3 4 3



3
3
3 3
2
2

3
x 3x
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3
x
3
x2
2

3
2 x
3
2
x3
3
2 x

x
2
Slide R-21
R.5 Rationalizing a Binomial
Denominator
Example Rationalize the denominator of
1
1 2
Solution


1 1 2
1
1 2


 1  2
1 2
1 2 1 2 1 2

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

Slide R-22