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!