Transcript Chapter 6
Chapter 6
Irrational and
Complex Numbers
Section 6-1
Roots of Real
Numbers
Square Root
A square root of a
number b is a solution
of the equation x2 = b.
Every positive number b
has two square roots,
denoted √b and -√b.
Principal Square Root
The positive square root
of b is the principal
square root
The principal square
root of 25 is 5
Examples – Square Root
Simplify
2
x
=9
x2 + 4 = 0
2
5x = 15
Cube Root
A cube root of b is a
solution of the equation
x3 = b.
Examples – Cube Root
Simplify
3
√8
3
√27
3√106
3
9
√a
nth root
1.
2.
3.
is the solution of xn = b
If n is even, there
could be two, one or no
nth root
If n is odd, there is
exactly one nth root
Examples – nth root
Simplify
4
√81
5
√32
5√-32
6
√-1
Radical
The symbol n√b is called
a radical
Each symbol has a
name
n = index
√ = radical
b = radicand
Section 6-2
Properties of
Radicals
Product and Quotient
Properties of Radicals
1. n√ab = n√a · n√b
2.
n√a÷b
=
n√a
÷
n√b
Examples
Simplify
3√25 · 3√10
3
√(81/8)
2
√2a b
3
√36w
Rationalizing the
Denominator
Create a perfect
square, cube or other
power in the
denominator in order
to simplify the answer
without a radical in
the denominator
Examples
Simplify
√(5/3)
4
3√c
Theorems
1. If each radical
represents a real
number, then
nq√b = n√(q√b).
2. If n√b represents a
real number, then
n√bm = (n√b)m
Examples
Give the decimal
approximation to the
nearest hundredth.
4√100
3√1702
Section 6-3
Sums of Radicals
Like Radicals
Two radicals with the
same index and same
radicand
You add and subtract
like radicals in the
same way you
combine like terms
Examples
Simplify
√8 + √98
3
3
√81 - √24
√32/3 + √2/3
Examples
Simplify
5
3
2
√12x - x√3x + 5x √3x
Answer
6x2√3x
Section 6-4
Binomials
Containing
Radicals
Multiplying Binomials
You multiply
binomials with
radicals just like you
would multiply any
binomials.
Use the FOIL method
to multiply binomials
Examples
Simplify
(4 + √7)(3 + 2√7)
Answer
26 + 11√7
Conjugate
Expressions of the
form a√b + c√d and
a√b - c√d
Conjugates can be
used to rationalize
denominators
Example - Conjugate
Simplify
3 + √5
3 - √5
Answer
7 + 3√5
2
Example - Conjugate
Simplify
1
4 - √15
Answer
4 + √15
Section 6-5
Equations
Containing
Radicals
Radical Equation
An equation which
contains a radical
with a variable in the
radicand.
40 = √22d
Solving a Radical
Equation
First isolate the
radical term on one
side of the equation
Solving a Radical
Equation - Continued
If the radical term is a
square root, square
both sides
If the radical term is a
cube root, cube both
sides
Example 1
Solve
√(2x – 1) = 3
Answer
X = 5
Example 2
Solve
23√x – 1 = 3
Answer
X = 8
Example 3
Solve
√(2x + 5) =2√2x + 1
Answer
X = 2/9
Section 6-6
Rational and
Irrational
Numbers
Completeness Property
of Real Numbers
Every real number
has a decimal
representation, and
every decimal
represents a real
number
Remember…
A rational number is
any number that can
be expressed as the
ratio or quotient of
two integers
Decimal Representation
Every rational number
can be represented by
a terminating decimal
or a repeating
decimal
Example 1
Write each
terminating decimal
as a fraction in lowest
terms.
2.571
0.0036
Example 2
Write each repeating
decimal as a fraction
in lowest terms.
0.32727…
1.89189189…
Remember…
An irrational number
is a real number that
is not rational
Decimal Representation
Every irrational number is
represented by an infinite
and nonrepeating decimal
Every infinite and
nonrepeating decimal
represents an irrational
number
Example 3
Classify each number as
either rational or irrational
√2
√4/9
2.0303…
2.030030003…
Section 6-7
The Imaginary
Number i
Definition
i = √-1
and
2
i
= -1
Definition
If r is a positive real
number, then
√-r = i√r
Example 1
Simplify
√-5
√-25
√-50
Combining imaginary
Numbers
Combine the same
way you combine like
terms
√-16 - √-49
i√2 + 3i√2
Multiply - Example
Simplify
√-4 ▪ √-25
i√2 ▪ i√3
Divide - Example
Simplify
2
3i
6
√-2
Example
Simplify
√-9x2 + √-x2
√-6y ▪ √-2y
Section 6-8
The Complex
Number
Complex Numbers
Real numbers and
imaginary numbers
together form the set of
complex numbers
The form a + bi,
represents a complex
number
Equality of Complex
Numbers
a + bi = c +di
if and only if
a = c and b = d
Sum of Complex
Numbers
(a + bi ) +(c +di ) =
(a + c) + (b + d)i
Product of Complex
Numbers
(a + bi )▪(c +di )=
(ac – bd) + (ad + bc)i
Example 1
Simplify
(3 + 6i) + (4 – 2i)
(3 + 6i) - (4 – 2i)
Example 2
Simplify
(3 + 4i)(5 + 2i)
2
4i)
(3 +
(3 + 4i)(3 - 4i)
Using Conjugates
Simplify using
conjugates
5–i
2 + 3i
Reciprocals
Find the reciprocal of
3–i
Remember…
the reciprocal of x = 1/x
THE END!