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Section 7.1
Rational Exponents
and Radicals
OBJECTIVES
A
Find the nth root of a
number, if it exists.
OBJECTIVES
B
Evaluate expressions
containing rational
exponents.
OBJECTIVES
C
Simplify expressions
involving rational
exponents.
DEFINITION
NTH ROOT
If a and x are real numbers
and n is a positive integer:
x is an nth root of a if xn = a
DEFINITION
PRINCIPLE NTH ROOT
If n is a positive integer, then
na
denotes the principle nth root of a
DEFINITION
RATIONAL EXPONENTS AND
THEIR ROOTS
If n is a positive integer and
n a is a real number:
a1/ n = n a
DEFINITION
RADICAL EXPRESSION WITH
AN M/N EXPONENT
m/ n
n
m
n m
a
=( a) = a
Provided m and n are
positive integers and n a
is a real number.
LAWS OF EXPONENTS
If r, s, and t are rational, and a
and b are real:
I. ar as = ar+s
r
a
r –s
II. s = a
a
LAWS OF EXPONENTS
If r, s, and t are rational, and a
and b are real:
III. (ar )s = ar
r s t
s
rt st
IV. (a b ) = a b
Chapter 7
Section 7.1A
Practice Test
Exercise #1
Find, if possible.
3
a.
=
3
–64
 – 4
= –4
3
Find, if possible.
b.
–36
It is not a real number.
Chapter 7
Section 7.1B
Practice Test
Exercise #4
Evaluate if possible.
a.
–27
–
2
3
=
=
=
1
2
 – 27  3
1



3

27
– 

1
 –3 
2
=
2
1
9
Evaluate, if possible.
b. 8
–2
3
1
=
8
=
2
3
1


3
1
=
22
2

8

=
1
4
Section 7.2
Simplifying Radicals
OBJECTIVES
A
Simplify radical
expressions.
OBJECTIVES
B
Rationalize the
denominator of a
fraction.
OBJECTIVES
C
Reduce the order of a
radical expression.
DEFINITION
nTH ROOT
1/n
a
=
n
a (a  0)
when n is a positive integer.
LAWS
For Simplifying Radical
Expressions
I.
n
n
a =a
(a  0)
LAWS
For Simplifying Radical
Expressions
II.
n
n
n
ab = a b Product rule (a, b  0)
III.
LAWS
For Simplifying Radical
Expressions
n
a
b
n
=
n
a
b
Quotient rule (a  0 ,b > 0)
DEFINITION
n an
n
n
a = |a |
n is an even positive integer
DEFINITION
n an
n
n
a =a
n is an odd positive integer
Chapter 7
Section 7.2A
Practice Test
Exercise #7b
Simplify.
4  – x 4
b.
= 4
–x
=
–x
=
x
4
Chapter 7
Section 7.2A
Practice Test
Exercise #8b
Simplify.
3
b.
=
=
54a 4 b12
3
3
27 • 2 • a3 • a • b12
33 a3 b12 • 2a
= 3ab
4 3
2a
Chapter 7
Section 7.2A
Practice Test
Exercise #9b
Simplify.
3
b.
5
x6
3
=
3
3
=
5
x6
5
x2
Section 7.3
Operations with
Radicals
OBJECTIVES
A
Add and subtract similar
radical expressions.
OBJECTIVES
B
Multiply and divide
radical expressions.
OBJECTIVES
C
Rationalize the
denominators of radical
expressions involving
sums or differences.
DEFINITION
LIKE RADICAL EXPRESSIONS
Radical expressions with the
same index and the same
radicand.
DEFINITION
CONJUGATE
The expressions a + b and
a – b are conjugates of each
other.
Chapter 7
Section 7.3A
Practice Test
Exercise #14
Perform the indicated operations.
a.
32 + 98
=
16 • 2 + 49 • 2
= 4 2 +7 2
= 11 2
Perform the indicated operations.
b.
112 –
=
28
16 • 7 –
= 4 7 – 2 7
= 2 7
4 •7
Chapter 7
Section 7.3B
Practice Test
Exercise #16b
Perform the indicated operations.
b.
3
=
=

3x 

3
9x –
3
3x •
3
3
–
3
8 • 6x
27x
= 3x –
= 3x – 2
3
2
3
3
9x
2
3
6x

16 x 

–
48x
2
2
3
2
3x •
3
16x
Chapter 7
Section 7.3B
Practice Test
Exercise #18b
Find the product.

b.
=
6 + 3  6 – 3 

6 + 3  6 – 3 
(a + b)(a – b) = a 2 – b 2
=
 6 – 3
2
= 6 – 3
= 3
2
Chapter 7
Section 7.3C
Practice Test
Exercise #20
Rationalize the denominator.
2
x –5
=
2
x –5
•
2 ( x + 5)
=
x – 25
2 x + 10
=
x – 25
x +5
x +5
Section 7.4
Solving Equations
Containing Radicals
OBJECTIVES
A
Solve equations
involving radicals.
OBJECTIVES
B
Solve applications
requiring the solution of
radical equations.
DEFINITION
POWER RULE OF EQUATIONS
All solutions of the equation
P = Q are solutions of the
n
n
equation P = Q , where n is
a natural number.
PROCEDURE
TO SOLVE EQUATIONS
CONTAINING RADICALS
• Isolate
• Repeat
• Raise
• Solve
• Simplify
• Check
Chapter 7
Section 7.4A
Practice Test
Exercise #21
Solve.
a.
x +2 = –2
There is no real number solution
because
x + 2 ° negative number.
When solving by squaring both sides:
x+2=4
x=2
However, the solution x = 2
does not check.
Solve.
b.

x = 14
x =7
x + 2 = x – 10
x +2

2
=
 x – 10 
2
2
x + 2 = x – 20x + 100
2
0 = x – 21x + 98
0 =
x – 14 = 0 or
 x – 14  x – 7 
x –7 =0
Solve.
b.
Check:
x = 7,
x = 14
x + 2 = x – 10
?
7 + 2 = 7 – 10
9 ° –3
x = 14,
?
14 + 2 = 14 – 10
16 = 4
The only solution is x = 14 .
x =7
Chapter 7
Section 7.4A
Practice Test
Exercise #22
Solve.
x –3 – x = –3
x –3 = x –3

x –3

2
=
 x – 3
2
x – 3 = x 2 – 6x + 9
2
–3 = x – 7x + 9
Solve.
x =4
x –3 – x = –3
2
–3 = x – 7x + 9
2
0 = x – 7x + 12
0 =
 x – 4  x – 3 
x – 4 = 0 or x – 3 = 0
x =3
Solve.
x =4
x =3
x –3 – x = –3
Check:
x = 4,
?
4–3 –4 = –3
–3 = – 3
x = 3,
?
3 – 3 –3 = – 3
–3 = – 3
The solutions are x = 4 or x = 3 .
Section 7.5
Complex Numbers
OBJECTIVES
A
Write the square root of
a negative integer in
terms of i.
OBJECTIVES
B
Add and subtract
complex numbers.
OBJECTIVES
C
Multiply and divide
complex numbers.
OBJECTIVES
D
Find powers of i.
DEFINITION
COMPLEX NUMBER
If a and b are real numbers, the
following is a complex number:
a + bi
 
Real part Imaginary part
RULES
ADDING AND SUBTRACTING
COMPLEX NUMBERS
(a + bi) + (c + di) = (a + c) + (b +d)i
(a + bi) – (c + di) = (a – c) + (b – d)i
PROCEDURE
DIVIDING ONE COMPLEX
NUMBER BY ANOTHER
Multiply the numerator and
the denominator by the
conjugate of the denominator.
Chapter 7
Section 7.5A
Practice Test
Exercise #26b
Write in terms of i.
b.
–98 = –1 • 49 • 2
=i •7 2
= 7i 2
Chapter 7
Section 7.5B
Practice Test
Exercise #27b
Find.
b.
 5 + 2i 
–  7 – 8i 
= 5 + 2i – 7 + 8i
= 5 – 7 + 2i + 8i
= – 2 + 10i
Chapter 7
Section 7.5C
Practice Test
Exercise #29b
Find.
b.
=
=
4 – 3i
3 – 5i
4
3
– 3i  3 + 5i 
– 5i  3 + 5i 
4  3  + 4  5i  +  –3i  3  +  –3i  5i 
 3
2
–  5i 
2
Find.
4 – 3i
3 – 5i
b.
4  3  + 4  5i  +  –3i  3  +  –3i  5i 
=
 3
=
2
–  5i 
12 + 20i – 9i – 15i
9 – 25i
2
2
2
Find.
4 – 3i
3 – 5i
b.
=
12 + 20i – 9i – 15i
9 – 25i
2
2
12 + 11i + 15
=
9 + 25
27 + 11i
=
34
27 11
=
+
i
34 34
Chapter 7
Section 7.5D
Practice Test
Exercise #30b
Write the answer as 1, -1, i or -i.
b. i – 9 =
1
9
i
=
1
8
i • i
=
1
1•i
1
=
=
=
i
i •i
1•i
=
i
i
2
= –i
i
=
–1
1
2
i  • i
4