# Chapter 7

```Intermediate Algebra
Chapter 7 - Gay
Oprah Winfrey
• “Although there may be tragedy in
possibility to triumph. It doesn’t
matter who you are, where you
come from. The ability to triumph
begins with you. Always.”
Angela Davis – U.S. political
activist-1987 – Spellman college
grasping things at the
root.”
Intermediate Algebra 7.1
Objective
• Find the nth root of a
number
Definition of nth root
• For any real numbers a and b and any
integer n>1, a is a nth root of b if and only if
a b
n
Principal nth root
• Even roots
• Principal nth root of b is the
nonnegative nth root of b.
• Represented by
n
b
n
n is index
Graphs determine domain & range
f ( x)  x
Graphs – determine domain &
range
g ( x)  x
3
Calculator keys




3

 MATH   4 : ( 
x

 MATH   
n
bn  b
If b is any real number
• For even integers
n
b b
n
If b is any real number
• For odd integers n
n
b b
n
Objectives
• 1. Find the nth root of a number
• 2. Approximate roots using
calculator.
• 4. Determine domain and range of
Intermediate Algebra 7.2
•Rational Exponents
Rational Exponent – numerator
of 1
• For any real number b for which the nth
roof of b is defined and any integer n>1
n
b b
1
n
Definition of
m
n
b 
b
 
n
b
m
m
n
 b
n
m
Problem
2
3
8 8
2
3
1
2
3
2
 
2
 8   2  4
 
1
3
8  [8][^][(2 / 3)][ ENTER]
Negative exponents
1
b  n
b
m
1
n
b  m
n
b
n
Rule & example
a
 
b
n
b
 
a
1
2
n
1
2
1
2
16
4
 25 
 16 



 
 
1
5
 16 
 25 
2
25
Althea Gibson – tennis player
• “No matter what
accomplishments you
make, someone helped
you.”
Intermediate Algebra 8.3
•Properties
•of
•Rational Exponents
Properties of exponents
a a a
m
n
m n
m
a
mn

a
n
a
a 
m n
a
mn
a 1 a  a
0
1
n
a
a

 
n
b
b
 ab 
n
n
a b
n n
Procedure: Reduce the Index
• 1. Write the radical in
exponential form
• 2. Reduce exponent to lowest
terms.
• 3. Write the exponential
Objectives:
• 1. Evaluate rational exponents.
• 2. Write radicals as expressions raised to
rational exponents.
• 3. Simplify expressions with rational
number exponents using the rules of
exponents.
Thomas Edison
• “I am not discouraged,
because every wrong
another step forward.”
Intermediate Algebra 7.3
•The Product Rule
•for
• For all real numbers a and b for which the
operations are defined
n
a
n
b
n
ab
the product.
Condition 1
simplified n-th root
perfect n-th power factor.
Using product rule to simplify
• 1. Write the radicand as a product of the
greatest possible perfect nth power and a
number that has no perfect nth power
factors.
• 2. Use product rule
• 3. Find the nth root of perfect nth power
• 4. Do all necessary simplifications
Sample problem
5 72  5 36 2 
5 36
30 2
2 5 6 2 
Sample Problem
5
64 x y  32  2  x  x  y  y
5
32 x y  2 x y 
9 12
5 10
5
5
4 2
2 xy 2 x y
5
5
4 2
4
10
2
Winston Churchill
• “I am an optimist.”
Intermediate Algebra 7.5
•The Quotient Rule
•for
• For all real numbers a and b for which the
operations are defined.
n
n
a
a

n
n
b
b
• The radical of a quotient is the quotient of
• The radicand of a simplified
fraction
7
9
3
9
3
x
Simplifying a radical – condition 3
• A simplified radical must not
denominator.
5
4
7
3
x
3
Rationalizing the denominator
• Square Roots
• 1. Multiply both the numerator
and denominator by the same
square root as appears in the
denominator.
• 2. Simplify.
Sample problem
5
5 6



6
6 6
30
30

6
36
Rationalizing a denominator
• Multiply the numerator and
denominator by the expression
that will make the radicand of
the denominator a perfect nth
power.
Example problem
3
2
3
3
2



3
3
3 2
2
2 2
3
3
3 4 3 4

3
2
8
Stanislaw J. Lec
• “He who limps is still
walking.”
Intermediate Algebra 8.6
• Operations
•with
Objective
• and
• * Identical indexes.
• To add or subtract like
coefficients and keep the
Procedure- multiplication with
•
•
•
•
Use Product Rule
Use distributive property
Use FOIL if needed
Conjugates
• A+B and A-B are called conjugates of each
other.
• Examples:


 5  3 
6  2  6  2 
5 3
Rationalizing a binomial
• Multiply the numerator and
denominator by the conjugate
of the denominator.
• Combine and Simplify
Rationalizing a binomial
• Multiply the numerator and
denominator by the conjugate
of the numerator.
• Combine and Simplify
Objective
• Rationalize binomial
denominator involving
Lance Armstrong
• “I didn’t just jump back on
the bike and win. There
were a lot of ups and downs,
but this time I didn’t let the
lows get to me.”
Intermediate Algebra 7.7
•Complex
•Numbers
Definition: imaginary number i
• The symbol I represents an imaginary
number with the following properties:
i  1 and
i  1
2
Definition
• For any positive real number n
n  i n
Definition: Complex Number
• A number that can be
expression the form
• a + bi where a and b are
real numbers and i is the
imaginary unit.
a+bi
•
•
•
•
•
a is called the real part
b is called the imaginary part
a+bi is standard form
a+0i is a real number = a
0 + bi =bi is pure imaginary
number
Set of Complex Numbers
• Set of Real numbers = R union
with set of Imaginary numbers
= I is the set of Complex
numbers=C
R
I C
Equality of Complex Numbers
• a + bi = c + di if and only if
• a = b and c = d
• Real parts are equal and
imaginary parts are equal
• (a+bi)+(c+di) = (a + c) + (b + d)i
• (a+bi) - (c+di) = (a - c)+(b – d)I
• Add or subtract the real and imaginary
parts.
Multiplication of complex
numbers
•
•
•
•
•
Translation:
1. Use FOIL
2
i  1
2. Substitute
3. Combine terms
4. Write in standard form
2i  35i  10i
49  4i 2
47
 i
53
Division of imaginary number by
real number
• To divide a + bi by a nonzero
real number c, divide real part
and imaginary part by c.
a  bi a b
  i
c
c c
Division by Complex Numbers
• 1. Multiply numerator and
denominator by complex
conjugate of denominator.
• 2. Combine and simplify
• 3. *** Write in standard form.
Sample Problem
6  5i 6  5i 7  2i



7  2i 7  2i 7  2i
42  12i  35i  10i
2
49  4i
2
32 47

 i
53 53
George Simmel - Sociologist
• “He is educated who
knows how to find out
what he doesn’t know.”
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