Download 7.2

Chapter 7
Roots,
Radicals, and
Complex
Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-1
1
Chapter Sections
7.1 – Roots and Radicals
7.2 – Rational Exponents
7.3 – Simplifying Radicals
7.4 – Adding, Subtracting, and Multiplying
Radicals
7.5 – Dividing Radicals
7.6 – Solving Radical Equations
7.7 – Complex Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-2
2
§ 7.2
Rational Exponents
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-3
3
Changing a Radical Expression
A radical expression can be written using exponents
by using the following procedure:
n
a a
1
n
When a is nonnegative, n can be any index.
When a is negative, n must be odd.
7 7
3
7
1
2
15ab  15ab 
13
 4 x y   4 x y
2
5
2

5 17
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-4
4
Simplifying Radical Expressions
This rule can be expanded so that radicals of the
form n a m can be written as exponential expressions.
For any nonnegative number a, and integers m and n,
Power
n
8x
a 
m
 a
n
m
 am n
Index
b2 3  3 b2
2
 9y

73

 8x
3
2
9y

Copyright © 2015, 2011, 2007 Pearson Education, Inc.
7
Chapter 7-5
5
Simplify Radical Expressions
Exponential form of
n
an
For any nonnegative number a,
n
a  a
n
n
n

n
 an/ n  a
Examples
32  3
6
( xy ) 6  xy
4
y y
4
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-6
6
Rules of Exponents
The rules of exponents from Section 5.1 also apply
when the exponents are rational numbers.
For all real numbers a and b and all rational numbers m and n,
Product rule:
am • an = am + n
Quotient rule:
Negative exponent rule:
am
mn

a
, a0
n
a
1
m
a  m , a0
a
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-7
7
Rules of Exponents
For all real numbers a and b and all rational numbers m and n,
Zero exponent rule:
Raising a power to a power:
a0 = 1, a 0
a 
m n
 a m n
m
m m


ab

a
b
Raising a product to a power :
m
am
a
Raising a quotient to a power :  b   b m , b  0
 
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-8
8
Rules of Exponents
Examples:
1.) Evaluate 8-2/3.
8
2 / 3
1
 2/3
8
1
 3 2
 8
1 1
 2
2 4
2.) Evaluate
(27) 5 / 3 
1
1
1
1




5
(27) 5 / 3 3  27  (3) 5
243
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-9
9
Rules of Exponents
Examples Simplify each expression and write the answer
without negative exponents.
1.)
a1 / 2  a 2 / 3  a (1 / 2 )  ( 2 / 3 )
a
2.)
1 / 6
1
 1/ 6
a
(6 x 2 y 4 ) 1 / 2  6 1 / 2 x 2 ( 1 / 2 ) y 4 ( 1 / 2 )
 6 1 / 2 x 1 y 2

y2
1/ 2
6 x
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-10
10