Download Chapter 7

Chapter 7
Rational Exponents,
Radicals, and
Complex Numbers
§ 7.1
Radicals and Radical
Functions
Square Roots
Opposite of squaring a number is taking the
square root of a number.
A number b is a square root of a number a if
b2 = a.
In order to find a square root of a, you need a
# that, when squared, equals a.
Martin-Gay, Intermediate Algebra, 5ed
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Principal Square Roots
Principal and Negative Square Roots
If a is a nonnegative number, then
a is the principal or nonnegative square
root of a
 a is the negative square root of a.
Martin-Gay, Intermediate Algebra, 5ed
4
Radicands
Radical expression is an expression
containing a radical sign.
Radicand is the expression under a radical
sign.
Note that if the radicand of a square root is a
negative number, the radical is NOT a real
number.
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Radicands
Example:
49 
7
5
25

16
4
 4  2
Martin-Gay, Intermediate Algebra, 5ed
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Perfect Squares
Square roots of perfect square radicands
simplify to rational numbers (numbers that
can be written as a quotient of integers).
Square roots of numbers that are not perfect
squares (like 7, 10, etc.) are irrational
numbers.
IF REQUESTED, you can find a decimal
approximation for these irrational numbers.
Otherwise, leave them in radical form.
Martin-Gay, Intermediate Algebra, 5ed
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Perfect Square Roots
Radicands might also contain variables and
powers of variables.
To avoid negative radicands, assume for this
chapter that if a variable appears in the
radicand, it represents positive numbers only.
Example:
64x10  8x 5
Martin-Gay, Intermediate Algebra, 5ed
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Cube Roots
Cube Root
The cube root of a real number a is written as
3
a, and
3
a  b only if b 3  a
Martin-Gay, Intermediate Algebra, 5ed
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Cube Roots
Example:
3
27  3
3
 8x 6   2x 2
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nth Roots
Other roots can be found, as well.
The nth root of a is defined as
n
a  b only if b n  a
If the index, n, is even, the root is NOT a
real number when a is negative.
If the index is odd, the root will be a real
number.
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nth Roots
Example:
Simplify the following.
2 20
25a b
10
5ab

3
4
a
64
a
3
  3
9
b
b
Martin-Gay, Intermediate Algebra, 5ed
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nth Roots
Example:
Simplify the following. Assume that all
variables represent positive numbers.
4
2
16x8  2x
5
2a
32
a
 2a 
5
  15     3   3
b
b
 b 
Martin-Gay, Intermediate Algebra, 5ed
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nth Roots
n
If the index of the root a is even, then the
notation represents a positive number.
But we may not know whether the variable a is
a positive or negative value.
Since the positive square root must indeed be
positive, we might have to use absolute value
signs to guarantee the answer is positive.
Martin-Gay, Intermediate Algebra, 5ed
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Finding nth Roots
n
Finding a
n
If n is an even positive integer, then
n
a a
n
If n is an odd positive integer, then
n
a a
n
Martin-Gay, Intermediate Algebra, 5ed
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Finding nth Roots
Simplify the following.
2 20
25a b
 5ab
10
 5 a b10
If we know for sure that the variables represent
positive numbers, we can write our result
without the absolute value sign.
2 20
25a b
 5ab
10
Martin-Gay, Intermediate Algebra, 5ed
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Finding nth Roots
Example:
Simplify the following.
3
4a
64
a
3

  3
9
b
b
Since the index is odd, we don’t have to force
the negative root to be a negative number.
If a or b is negative (and thus changes the sign
of the answer), that’s okay.
Martin-Gay, Intermediate Algebra, 5ed
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Evaluating Rational Functions
We can also use function notation to represent rational
functions.
For example, T (x)  2  x  4.
x
Evaluating a rational function for a particular value
involves replacing the value for the variable(s) involved.
Example:
2 x
 4.
Find the value T (2) 
x
22
T (2) 
4
2
4
  4  2  4  2
2
Martin-Gay, Intermediate Algebra, 5ed
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Root Functions
Since every value of x that is substituted into the
equation
yn x
produces a unique value of y, the root relation
actually represents a function.
The domain of the root function when the index
is even, is all nonnegative numbers.
The domain of the root function when the index
is odd, is the set of all real numbers.
Martin-Gay, Intermediate Algebra, 5ed
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Root Functions
We have previously worked with graphing basic
forms of functions so that you have some familiarity
with their general shape.
You should have a basic familiarity with root
functions, as well.
Martin-Gay, Intermediate Algebra, 5ed
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Graphs of Root Functions
Example:
y
Graph y  x
x
y
6
6
4
2
2
1
2
1
0
0
(6, 6 )
(4,
2)
(2, 2 )
(1, 1)
x
(0, 0)
Martin-Gay, Intermediate Algebra, 5ed
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Graphs of Root Functions
Example: Graph y  3 x
x
y
8
2
4
1
0
3
4
3
(4, 4 )
(-1, -1)
0
-1
-4
3 4
-2
(8, 2)
(1, 1)
1
-1
-8
y
(-8, -2)
(0, 0)
x
(-4,  3 4)
Martin-Gay, Intermediate Algebra, 5ed
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§ 7.2
Rational Exponents
Martin-Gay, Intermediate Algebra, 5ed
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Exponents with Rational Numbers
So far, we have only worked with integer
exponents.
In this section, we extend exponents to
rational numbers as a shorthand notation when
using radicals.
The same rules for working with exponents
will still apply.
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Understanding
1/n
a
Recall that a cube root is defined so that
3
a  b only if b 3  a
However, if we let b = a1/3, then
b3  (a1/ 3 )3  a 1/ 33  a1  a
Since both values of b give us the same a,
a1/ 3  3 a
Definition of a
1
n
If n is a positive integer greater than 1 and n a is a
real number, then
a1/ n  n a
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Using Radical Notation
Example:
Use radical notation to write the following.
Simplify if possible.
811/ 4  4 81  4 34  3
32x 
10 1/ 5
16x 
7 1/ 3

3
 5 32 x10  5 25 x10  2 x 2
16 x  2 x  2 x  2 x  2 x  2 x
7
3
4
7
3
3
6
Martin-Gay, Intermediate Algebra, 5ed
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2 3
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Understanding
m/n
a
1
Definition of a n
If m and n are positive integers greater than 1
with m/n in lowest terms, then
a
m/n
 a 
n
m
 a
n
m
as long as n a is a real number
Martin-Gay, Intermediate Algebra, 5ed
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Using Radical Notation
Example:
Use radical notation to write the following.
Simplify if possible.
8
4/3
 
 8
3
4
 
 2    2  16
3
3
4
4
3x  7 7 / 3  3 3x  77  3 3x  76  3 3x  7 
(3x  7)  3x  7
2 3
Martin-Gay, Intermediate Algebra, 5ed
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Understanding
Definition of a
a
-m

m/n
a
n
m / n

1
a
m/n
as long as a-m/n is a nonzero real number.
Martin-Gay, Intermediate Algebra, 5ed
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Using Radical Notation
Example:
Use radical notation to write the following.
Simplify if possible.
64
 16
2 / 3
5 / 4
1


2/3
64
1
1

 64   4 
2
3
1
  5/ 4  
16
3
1
2
4
4
5
3
2
1
1
 2 
16
4
1
1
  5 
2
32
Martin-Gay, Intermediate Algebra, 5ed
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Using Rules for Exponents
Example:
Use properties of exponents to simplify the
following. Write results with only positive
exponents.
32
1/ 5
x

2/3 3
 32
3/ 5
x 
2
 2  x
5
5
3
2
3
2
2
2

x

8x

a1/ 4  a 1/ 2
1

1/ 4 1/ 2  2 / 3 

3 / 12 6 / 12 8 / 12 
11/ 12
 a
a
a
 11/12
2/3
a
a
Martin-Gay, Intermediate Algebra, 5ed
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Using Rational Exponents
Example:
Use rational exponents to write as a single
radical.
3
5 2  5 2
1/ 3
1/ 2
 5
2/6
2
3/ 6

 5 2
2
Martin-Gay, Intermediate Algebra, 5ed

3 1/ 6

6
200
32
§ 7.3
Simplifying Radical
Expressions
Martin-Gay, Intermediate Algebra, 5ed
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Product Rule for Radicals
Product Rule for Radicals
If n a and n b are real numbers, then
n
a  b  ab
n
n
Martin-Gay, Intermediate Algebra, 5ed
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Simplifying Radicals
Example:
Simplify the following radical expressions.
40 
4  10  2 10
5

16
5
5

4
16
15
No perfect square factor, so the
radical is already simplified.
Martin-Gay, Intermediate Algebra, 5ed
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Simplifying Radicals
Example:
Simplify the following radical expressions.
x 
6
x x 
x  x x
20

16
x
20
4 5
2 5

8
x
x8
7
x16

6
Martin-Gay, Intermediate Algebra, 5ed
3
x
36
Quotient Rule Radicals
Quotient Rule for Radicals
If n a and n b are real numbers,
and n b is not zero, then
n
n
a
a
n
b
b
Martin-Gay, Intermediate Algebra, 5ed
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Simplifying Radicals
Example:
Simplify the following radical expressions.
3
3
16  3 8  2 
3

64
3
3
3

64
3
3
8 3 2  2 3 2
3
4
Martin-Gay, Intermediate Algebra, 5ed
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The Distance Formula
Distance Formula
The distance d between two points (x1,y1) and (x2,y2)
is given by
d
x2  x1    y2  y1 
2
Martin-Gay, Intermediate Algebra, 5ed
2
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The Distance Formula
Example:
Find the distance between (5, 8) and (2, 2).
d
x2  x1    y2  y1 
d
 5  (2)  8  2
d
 3  6
2
2
2
2
2
2
d  9  36  45  3 5
Martin-Gay, Intermediate Algebra, 5ed
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The Midpoint Formula
Midpoint Formula
The midpoint of the line segment whose
endpoints are (x1,y1) and (x2,y2) is the point with
coordinates
 x1  x2 , y1  y2 
 2

2


Martin-Gay, Intermediate Algebra, 5ed
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The Midpoint Formula
Example:
Find the midpoint of the line segment that
joins points P(5, 8) and P(2, 2).


 x1  x2 , y1  y2   (5)  (2) , 8  2
 2

2
2
2


7 10

,
2 2
  3.5,5

Martin-Gay, Intermediate Algebra, 5ed

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§ 7.4
Adding, Subtracting, and
Multiplying Radical
Expressions
Martin-Gay, Intermediate Algebra, 5ed
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Sums and Differences
Rules in the previous section allowed us to
split radicals that had a radicand which was a
product or a quotient.
We can NOT split sums or differences.
ab  a  b
a b  a  b
Martin-Gay, Intermediate Algebra, 5ed
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Like Radicals
In previous chapters, we’ve discussed the concept of “like”
terms.
These are terms with the same variables raised to the same
powers.
They can be combined through addition and subtraction.
Similarly, we can work with the concept of “like” radicals to
combine radicals with the same radicand.
Like radicals are radicals with the same index and the same
radicand.
Like radicals can also be combined with addition or
subtraction by using the distributive property.
Martin-Gay, Intermediate Algebra, 5ed
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Adding and Subtracting Radical Expressions
Example:
37 3  8 3
10 2  4 2  6 2
3
2 4 2
Can not simplify
5 3
Can not simplify
Martin-Gay, Intermediate Algebra, 5ed
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Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression.
 75  12  3 3 
 25  3  4  3  3 3 
 25  3  4  3  3 3 
5 3  2 3 3 3 
 5  2  3
Martin-Gay, Intermediate Algebra, 5ed
3  6 3
47
Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression.
3
64  3 14  9 
4  3 14  9   5  3 14
Martin-Gay, Intermediate Algebra, 5ed
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Adding and Subtracting Radical Expressions
Example:
Simplify the following radical expression. Assume
that variables represent positive real numbers.
3 45x  x 5x  3 9 x  5x  x 5x 
3
2
3 9 x 2  5x  x 5x 
3  3x 5 x  x 5 x 
9 x 5x  x 5x 
9 x  x 
5x 
Martin-Gay, Intermediate Algebra, 5ed
10 x 5 x
49
Multiplying and Dividing Radical Expressions
If
n
a and n b are real numbers,
n
a  n b  n ab
n
a n a

if b  0
b
b
n
Martin-Gay, Intermediate Algebra, 5ed
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Multiplying and Dividing Radical Expressions
Example:
Simplify the following radical expressions.
3 y  5x  15 xy
7 6
ab
3 2
ab

7 6
ab

3 2
ab
ab  ab
4 4
Martin-Gay, Intermediate Algebra, 5ed
2 2
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§ 7.5
Rationalizing Numerators
and Denominators of
Radical Expressions
Martin-Gay, Intermediate Algebra, 5ed
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Rationalizing the Denominator
Many times it is helpful to rewrite a radical quotient
with the radical confined to ONLY the numerator.
If we rewrite the expression so that there is no
radical in the denominator, it is called rationalizing
the denominator.
This process involves multiplying the quotient by a
form of 1 that will eliminate the radical in the
denominator.
Martin-Gay, Intermediate Algebra, 5ed
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Rationalizing the Denominator
Example:
Rationalize the denominator.
3
2


2
2
6
3 2

2
2 2
3
6 33
63 3
63 3
6 3
3




 2 3
3
3
3
3
3
3
27
3
9 3
9
Martin-Gay, Intermediate Algebra, 5ed
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Conjugates
Many rational quotients have a sum or
difference of terms in a denominator, rather
than a single radical.
In that case, we need to multiply by the
conjugate of the numerator or denominator
(which ever one we are rationalizing).
The conjugate uses the same terms, but the
opposite operation (+ or ).
Martin-Gay, Intermediate Algebra, 5ed
55
Rationalizing the Denominator
Example:
Rationalize the denominator.
2 3
3  2 3 2 2  2 3
32



2  3 2 2  3  2  3 3
2 3
6 3 2 2  2 3

23
6 3 2 2  2 3

1
 6 3 2 2  2 3
Martin-Gay, Intermediate Algebra, 5ed
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Rationalizing the Numerator
An expression rewritten with no radical in the
numerator is called rationalizing the
numerator.
Example:
12

7
12 12


7
12
12
12  12


84
7  12
12
12


4  21
2  21
Martin-Gay, Intermediate Algebra, 5ed
12

4  21
6
21
57
Rationalizing the Numerator
Example: Rationalize the numerator.
3
9y

7
3
3
x y
x y
9y
7


3
3y2
3
3y2

x y
x y
3
3

9 y  3 3y2
7  3 3y2

3
27 y 3
3
21 y 2
3y

3
21 y 2
x x y  x y  y
x x y  x y  y

x y

x  2 xy  y
x  xy  xy  y
x  xy  xy  y
Martin-Gay, Intermediate Algebra, 5ed
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§ 7.6
Radical Equations and
Problem Solving
Martin-Gay, Intermediate Algebra, 5ed
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The Power Rule
Power Rule
If both sides of an equation are raised to the same
power, solutions of the new equation contain all the
solutions of the original equation, but might also
contain additional solutions.
A proposed solution of the new equation that is NOT
a solution of the original equation is an extraneous
solution.
Martin-Gay, Intermediate Algebra, 5ed
60
Solving Radical Equations
Solving a Radical Equation
1) Isolate one radical on one side of the equation.
2) Raise each side of the equation to a power equal
to the index of the radical and simplify.
3) If the equation still contains a radical term, repeat
Steps 1 and 2. If not, solve the equation.
4) Check all proposed solutions in the original
equation.
Martin-Gay, Intermediate Algebra, 5ed
61
Solving Radical Equations
Example:
Solve the following radical equation.
x 1 1  0

x 1  1
Substitute into the
original equation.
x  1  12

2 1 1  0
x 1 1
1 1  0
2
x2
1 1  0 true
So the solution is x = 2.
Martin-Gay, Intermediate Algebra, 5ed
62
Solving Radical Equations
Example:
Solve the following radical equation.
2x  x 1  8

x 1  8  2x

x  1  8  2 x 
2
2
x  1  64  32 x  4 x 2
0  63  33x  4 x 2
0  (3  x)( 21  4 x)
21
x  3 or
4
Martin-Gay, Intermediate Algebra, 5ed
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Solving Radical Equations
Example continued:
Substitute the value for x into the original equation, to
check the solution.
2(3)  3  1  8
6  4  8 true
So the solution is x = 3.
 
21
21
2

1  8
4
4
21
25

8
2
4
21 5
 8
2 2
26
8
2
Martin-Gay, Intermediate Algebra, 5ed
false
64
Solving Radical Equations
Example:
Solve the following radical equation.
y 5  2 y 4

 
2
y 5  2 y 4

2
y 5  44 y 4  y 4
5  4 y  4
5
  y4
4
2
 5
  
 4

y4
25
 y4
16
25 89
y  4

16 16

2
Martin-Gay, Intermediate Algebra, 5ed
65
Solving Radical Equations
Example continued:
Substitute the value for x into the original equation, to
check the solution.
89
89
5  2
4
16
16
169
25
 2
16
16
13
5
 2
4
4
13 3

4 4
false
So the solution is .
Martin-Gay, Intermediate Algebra, 5ed
66
Solving Radical Equations
Example:
Solve the following radical equation.
2 x  4  3x  4  2

2 x  4  2  3x  4
 
2
2 x  4   2  3x  4

2
2 x  4  4  4 3x  4  3x  4
2 x  4  8  3x  4 3x  4
x 2  24 x  80  0
 x  12  4 3x  4
x  20x  4  0
 x  12


2
  4 3x  4
x 2  24 x  144  16(3x  4)  48x  64
2
Martin-Gay, Intermediate Algebra, 5ed
x  4 or 20
67
Solving Radical Equations
Example continued:
Substitute the value for x into the original
equation, to check the solution.
2(4)  4  3(4)  4  2
2(20)  4  3(20)  4  2
4  16  2
36  64  2
2  4  2
6  8  2
true
true
So the solution is x = 4 or 20.
Martin-Gay, Intermediate Algebra, 5ed
68
Solving Radical Equations
Example:
Solve the following radical equation.

x 1  5

2
x 1  5
2
x  1  25
Substitute into the
original equation.
24  1  5
x  24
25  5
true
So the solution is x = 24.
Martin-Gay, Intermediate Algebra, 5ed
69
Solving Radical Equations
Example:
Solve the following radical equation.
Substitute into the
5x  5
original equation.
 5x 
2
  5
5x  25
2
5  5  5
25  5
Does NOT check, since the left side
of the equation is asking for the
x 5
principal square root.
So the solution is .
Martin-Gay, Intermediate Algebra, 5ed
70
The Pythagorean Theorem
Pythagorean Theorem
If a and b are the lengths of the legs of a right
triangle and c is the length of the hypotenuse,
then
a2 + b2 = c2
There are several applications in this section
that require the use of the Pythagorean
Theorem in order to solve.
Martin-Gay, Intermediate Algebra, 5ed
71
Using the Pythagorean Theorem
Example:
Find the length of the hypotenuse of a right
triangle when the length of the two legs are
2 inches and 7 inches.
c2 = 22 + 72 = 4 + 49 = 53
c=
53 inches
Martin-Gay, Intermediate Algebra, 5ed
72
§ 7.7
Complex Numbers
Martin-Gay, Intermediate Algebra, 5ed
73
Imaginary Numbers
Previously, when we encountered square roots
of negative numbers in solving equations, we
would say “no real solution” or “not a real
number”.
Imaginary Unit
The imaginary unit i, is the number whose square is – 1.
That is,
i 2  1 and i  1
Martin-Gay, Intermediate Algebra, 5ed
74
The Imaginary Unit, i
Example:
Write the following with the i notation.
 25 
25   1  5 i
 32 
32   1  16  2   1  4 2  i  4 i 2


 121   121   1   11i
Martin-Gay, Intermediate Algebra, 5ed
75
Complex Numbers
Real numbers and imaginary numbers are both
subsets of a new set of numbers.
Complex Numbers
A complex number is a number that can be
written in the form a + bi, where a and b are real
numbers.
Martin-Gay, Intermediate Algebra, 5ed
76
Standard Form of Complex Numbers
Complex numbers can be written in the form
a + bi (called standard form), with both a and
b as real numbers.
a is a real number and bi would be an
imaginary number.
If b = 0, a + bi is a real number.
If a = 0, a + bi is an imaginary number.
Martin-Gay, Intermediate Algebra, 5ed
77
Standard Form of Complex Numbers
Example:
Write each of the following in the form of a
complex number in standard form a + bi.
6 = 6 + 0i
8i = 0 + 8i
 24 
4  6  1  2 i 6  0  2 i 6
6   25  6  25   1  6 + 5i
Martin-Gay, Intermediate Algebra, 5ed
78
Adding and Subtracting Complex Numbers
Sum or Difference of Complex Numbers
If a + bi and c + di are complex numbers, then
their sum is
(a + bi) + (c + di) = (a + c) + (b + d)i
Their difference is
(a + bi) – (c + di) = (a – c) + (b – d)i
Martin-Gay, Intermediate Algebra, 5ed
79
Adding and Subtracting Complex Numbers
Example:
Add or subtract the following complex numbers.
Write the answer in standard form a + bi.
(4 + 6i) + (3 – 2i) = (4 + 3) + (6 – 2)i = 7 + 4i
(8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i
Martin-Gay, Intermediate Algebra, 5ed
80
Multiplying Complex Numbers
The technique for multiplying complex
numbers varies depending on whether the
numbers are written as single term (either the
real or imaginary component is missing) or
two terms.
Martin-Gay, Intermediate Algebra, 5ed
81
Multiplying Complex Numbers
Note that the product rule for radicals does
NOT apply for imaginary numbers.
2
 16   25  4i  5i  20i  20(1)   20
 16   25 
16  25 
Martin-Gay, Intermediate Algebra, 5ed
400  20
82
Multiplying Complex Numbers
Example:
Multiply the following complex numbers.
8i · 7i
56i2
56(1)
56
Martin-Gay, Intermediate Algebra, 5ed
83
Multiplying Complex Numbers
Example:
Multiply the following complex numbers.
Write the answer in standard form a + bi.
5i(4 – 7i)
20i – 35i2
20i – 35(–1)
20i + 35
35 + 20i
Martin-Gay, Intermediate Algebra, 5ed
84
Multiplying Complex Numbers
Example:
Multiply the following complex numbers.
Write the answer in standard form a + bi.
(6 – 3i)(7 + 4i)
42 + 24i – 21i – 12i2
42 + 3i – 12(–1)
42 + 3i + 12
54 + 3i
Martin-Gay, Intermediate Algebra, 5ed
85
Complex Conjugate
Complex Conjugates
The complex numbers (a + bi) and (a – bi) are
complex conjugates of each other, and
(a + bi)(a – bi) = a2 + b2
Martin-Gay, Intermediate Algebra, 5ed
86
Complex Conjugate
The conjugate of a + bi is a – bi.
The conjugate of a – bi is a + bi.
The product of (a + bi) and (a – bi) is
(a + bi)(a – bi)
a2 – abi + abi – b2i2
a2 – b2(–1)
a2 + b2, which is a real number.
Martin-Gay, Intermediate Algebra, 5ed
87
Dividing Complex Numbers
Example:
Use complex conjugates to divide the following
complex numbers. Write the answer in standard
form.
24  18i  8i  6i 2
6  2i 6  2i 4  3i




2
4  3i 4  3i 4  3i 16  12i  12i  9i
18 26
24  26i  6(1) 18  26i

 i

25
25 25
16  9(1)
Martin-Gay, Intermediate Algebra, 5ed
88
Dividing Complex Numbers
Example:
Divide the following complex numbers.
5  6i
 30i
 30i
5
5
 30i





i

2
6i  6i  36i
36
6i
6
 36(1)
Martin-Gay, Intermediate Algebra, 5ed
89
Patterns of i
i  1
i  i  i  (1)i  i
i 2  1
i 6  i 4  i 2  (1)(1)  1
i  i  i  (1)i  i
4
2
2
i  i  i  (1)(1)  1
i  i  i  (1)(i)  i
8
4
4
i  i  i  (1)(1)  1
3
5
2
7
4
4
3
The powers recycle through each multiple of 4.
i
4k
1
Martin-Gay, Intermediate Algebra, 5ed
90
Patterns of i
Example:
Simplify each of the following powers.
i  i i  i
53
i
17
52
413
 i  1i  i
1
1
1 1 i i
1
  
 2 
 17  16 
i
i i 1 i i i  i  i
i
i
 i

1
 (1)
Martin-Gay, Intermediate Algebra, 5ed
91