Transcript Intermediate Algebra Chapter 10

Intermediate Algebra 098B
Chapter 10 - Gay
•Rational Exponents,
Radical Expressions
and Complex
Numbers
Oprah Winfrey
• “Although there may be tragedy in
your life, there’s always to
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
• “Radical simply means
grasping things at the
root.”
Intermediate Algebra 10.1
•Radicals
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
b is radical
n is index
b is radicand
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.
• 3. Graph radical functions
• 4. Determine domain and range of
radical functions.
• 5. Simplify radical expressions.
Intermediate Algebra 10.2
•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 098B
•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
expression as a radical.
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.
• 4. Simplify radical expressions
Thomas Edison
• “I am not discouraged,
because every wrong
attempt discarded is
another step forward.”
Intermediate Algebra 098B 10.3
•Simplifying Radical
Expressions
Product Rule for Radicals
• For all real numbers a and b for which the
operations are defined
n
a
n
b
n
ab
• The product of the radicals is the radical of
the product.
Simplifying a Radical
Condition 1
• The radicand of a
simplified n-th root
radical must not contain a
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
radicand.
• 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 098B
•The Quotient Rule
•for
•Radicals
Quotient Rule for Radicals
• 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 radical.
Simplifying a radical: condition 2
• The radicand of a simplified
radical must not contain a
fraction
7
9
3
9
3
x
Simplifying a radical – condition 3
• A simplified radical must not
contain a radical in the
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
containing a higher-order radical.
• 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 098B –
10.4
•Adding, Subtracting
and Multiplying
Radical Expressions
Objective
• Add or subtract like
radicals
Definition: Like Radicals
• Are radical expressions
• * with identical radicands
• and
• * Identical indexes.
Procedure – Adding like radicals
• Simplify all radicals first.
• To add or subtract like
radicals, add or subtract the
coefficients and keep the
radicals the same.
Procedure- multiplication with
radicals
•
•
•
•
Simplify all radicals first
Use Product Rule
Use distributive property
Use FOIL if needed
Intermediate Algebra 098B –
10.5
• Rationalizing Denominators
• and
• Numerators of Radical Expressions
Conjugates
• A+B and A-B are called conjugates of each
other.
• Examples:


 5  3 
6  2  6  2 
5 3
Rationalizing a binomial
denominator with radicals
• Multiply the numerator and
denominator by the conjugate
of the denominator.
• Combine and Simplify
• Denominator cannot be radical
Rationalizing a binomial
numerator with radicals
• Multiply the numerator and
denominator by the conjugate
of the numerator.
• Combine and Simplify
• Denominator cannot be radical
Objective
• Rationalize binomial
denominator involving
radicals.
Lance Armstrong
• “I didn’t just jump back on
the bike and win. There
were a lot of ups and downs,
good results and bad results,
but this time I didn’t let the
lows get to me.”
Intermediate Algebra 098B –
10.6
• Radical Equations
• and
• Problem Solving
Intermediate Algebra 098B –
10.7
•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
Add and subtract Complex #s
• (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
• (a+bi)(c+di)=(ac-bd) + (bc+ad)I
•
•
•
•
•
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.”