MATH 0322 Intermediate Algebra
Unit 2
Radical Expressions
and Functions
Section: 10.1
Radical Expressions and Functions
 Radical expressions contain a radical sign 𝑛 ( ).
• Match the name with the correct part of
the radical expression:
radical
𝑛
𝑎
radicand
index
• Square Root when 𝑛 is 2: 2 ( ) → ( )
3
• Cube Root when 𝑛 is 3:
( ) ….and so on.
𝑛
Depending on the _____,
( ) can either be
index
an even root or an odd root.
Radical Expressions and Functions
Definition: The ________
Principal Square Root (p693)
𝑎 = ____
𝑏 ,
If 𝑏2 = 𝑎, then (___)
where 𝑎 and 𝑏 are nonnegative real numbers.
• For example:
6 2 = ____
36 inside the square root.
36 = 6 because (___)
1
9
=
1
3
because
1 2
3
=
1
____
9
inside the square root.
Radical Expressions and Functions
 − (𝑎) is “the negative of” square root 𝑎.
(Not the same as square root of negative 𝑎.)
• For example:
− 9 is read as “the negative of” 9 , so
− 9= − 9
Since 9 is a real number,
=− 3
= −3
P
then − 9 is also.
Radical Expressions and Functions
 …but, what about (−𝑎), not a real number
is it a real number also?
• Try to simplify using the definition of the
Principal Square Root.
?
O
O
only if −3
3 2 = ____.
−9 Is it?
?
= −3 only if −3 2 = ____.
−9 Is it?
−9 is not a real number, so in general…
−9 = 3
Radical Expressions and Functions
• Practice: Evaluate the following.
121 = 11
− 81 = −9
−4 𝒏𝒐𝒕 𝒂 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓
49
100
=
7
10
64 + 36 = 8 + 6 = 14
9 + 16 = 25 = 5
Radical Expressions and Functions
• Use your notes to explain the following:
“Why is −25 not a real number?”
In general, to produce real numbers,
even roots can only operate on
radicand values that are:
P greater than or equal to zero?
A.
B.
less than or equal to zero?
• This also applies to the function 𝑓 𝑥 = 𝑥 .
Radical Expressions and Functions
 Complete the table to plot the graph of 𝑓 𝑥 = 𝑥.
𝒙≥𝟎
𝒇 𝒙 = 𝒙
(𝒙, 𝒚)
0
𝑓 0 = 0=0
(0,0)
1
𝑓 1 = 1=1
(1,1)
𝑓 4 = 4=2
(4,2)
∞
5
2
3
4
−∞
6
7
8
9
∞
−∞
5
𝑓 9 = 9=3
(9,3)
9
Radical Expressions and Functions
 The domain of a square root function 𝑓 𝑥 = ( )
radicand be nonnegative.
requires that the ________
(Nonnegative means ≥ 0.)
 Practice: To find domain of 𝑓 𝑥 = 6𝑥 − 18
1) set the radicand ≥ 0
6𝑥 − 18 ≥ 0
2) solve for 𝑥
6𝑥 ≥ 18
𝑥≥3
 The domain written in Interval Notation is [3, ∞).
P
Radical Expressions and Functions
 Definition: The Cube Root of a number
3
𝑎 3 = ____
𝑏
𝑎 = 𝑏, means that (___)
real numbers have cube roots.
• All ____
3
3
3
since if 5 3 = 125.
125 = 5
−64 = −4 since if −4 3 = −64.
8
−
27
=
2
−
3
since if
 The domain of 𝑓 𝑥 =
3
2 3
−
3
=
8
− .
27
real numbers.
𝑥 is all ____
Radical Expressions and Functions
Suggested Practice: Form a table of squares
and cubes, then review for quicker recall of
square roots and cube roots in later sections.
𝑥
𝑥2
0
0
1
1
2
4
3
9
4
16
5
25
…
…
𝑥3
0
1
8
27
64
125
…
MATH 0322 Intermediate Algebra
Unit 2
Rational Exponents
Section: 10.2
Rational Exponents
Complete the Exponent Properties below
for your notes and Formula Sheet.(p.709)
 Properties(𝑎, 𝑏 are real and 𝑚, 𝑛 are rational exponents):
1)
2)
3)
𝑏𝑚
∙
𝑏𝑛
=_____________
𝑏𝑚
= _____________
𝑏𝑛
𝑏 𝑚 𝑛 = _____________
4)
𝑎𝑏
5)
𝑎 𝑛
𝑏𝑚
−
6) 𝑎
𝑛
𝑛
= _____________
Multiply like bases: keep base, add powers,
then simplify
Divide like bases: keep base, subtract powers,
then simplify
Power to power: keep base, multiply powers,
then simplify
Product to power: raise each factor to power,
then simplify
= _____________
Quotient to power: raise numerator and
denominator to power, then simplify
= _____________
Negative power: take reciprocal of base,
𝒎
𝒏
𝒂 must be nonzero
Rational Exponents
fraction form.
• Rational exponents: exponents in _______
4
5
1
2
2
3
Example: 9 , (−32) , (16𝑥 3 )
 Definition:
1
𝑛
𝑎 =
𝑛
𝑎
Radical notation
Rational Exponent
notation
• If index is even, radicand must be ≥ 0.
81 = 81P
(−25) = −25
P
O
• If index is odd, radicand can be any real number.
0 = 0P
32 = 32P
(−128) = −128P
1
6
6
1
3
3
0 =
0
1
4
4
1
5
5
1
2
1
7
7
Rational Exponents
• Practice: Use the previous definition to rewrite
in radical notation, then simplify.
1
𝑛
𝑎 =
1
4
81 =
4
𝑎
81 =
P
4
3∙3∙3∙3 =3
1
3
3
−64 =
1
5
5
7𝑥 𝑦 =
(−64) =
(7𝑥 4 𝑦)
𝑛
=
4
3
P
(−4) ∙ (−4) ∙ (−4) = −4
P
5
7∙𝑥∙𝑥∙𝑥∙𝑥∙𝑦
Rational Exponents
• Practice: Use the previous definition to only
rewrite in rational exponent notation.
𝑛
4
5
3
𝑎=𝑎
1
𝑛
21 = 21
𝑤3
6
1
4
1
3 5
=
𝑤
6
−5𝑥𝑦 =
P
P
−5𝑥𝑦
1
3
P
Rational Exponents
• Numerator of rational exponent can be > 1.
4
5
3
2
Example: 9 , (−32) ,
𝑚
𝑛
Definition: If 𝑎 is real and
reduced, then

𝑚
𝑛
𝑎 =
𝑚
𝑛
 andt 𝑎 =
𝑛
𝑛
𝑎
𝑎𝑚
𝑚
𝑚
𝑛
2
(16𝑥 3 )7
is positive and
This form will be easier to
work with.
Rational Exponents
• Practice: Evaluate using definitions only.
𝑚
𝑛
𝑎 =
No calculator.
𝑛
𝑚
𝑛
𝑎𝑚
3
5
𝑛
and 𝑎 =
Exp. Prop. #6
2
125
3
4
=
𝑚
P…?
= )3 P
32
= …?
= −27 P
3
5
3
3
3
−32768
(−32)
= −2
(−32)
= −8=
−32 =
55
(−32) =
−3
𝑎
2
33
−81 = −
𝟏𝟏 2
33
(−32)
𝟏𝟐𝟓
𝟏𝟐𝟓
3
4
3
𝟏
(11 ) == 1𝟓 (
81
1
𝟏
𝟏
𝟏𝟓𝟔𝟐𝟓 𝟐𝟓
2
=− 3
3
Rational Exponents
• Practice: Simplify using properties.
Property
1
2
3
2
Expo.
Prop. #1
6 ∙6 =
Expo.
Prop. #2
7
86
5
86
Expo.
Prop. #6
3
1
1 33
+ )
( 2+
2
6 22
7
1 35
− )
( 2+
2
=8
−
16
1
4
6
6
=
P
=8
= 8=2P
1 = P
12
(2)
=8
6
1 1
43
= ( )4 =
16
125𝑥 6 =
14
6( 2 )
1
(125𝑥
125𝑥6 ) 3
11
(2)
3
3
𝟏
(
)
𝟏𝟔
=
= 6(2) = 36
𝟏
𝟐
1
6
(2
)
3 6) 3
( (5)
1 25𝑥
5
𝑥
1 1
2 1
3
3∙ 3
= (5
5) 1( 𝑥5)633 ∙ 31
= 5𝑥 2
P
Rational Exponents
Practice and Complete
HW10.2
MATH 0322 Intermediate Algebra
Unit 2
Multiplying and Simplifying
Radical Expressions
Section: 10.3
Multiplying and Simplifying
Radical Expressions
• To multiply and simplify Radical Expressions,
you must learn to use:
1) Factoring
2) Product Rule for Radicals
• Compare the following:
a)
4 ∙ 25 = 100 = 10
4 ∙ 25 = 2
∙5
P
= 10
3
b)
3
3
8 ∙ 27 =
3
216 = 6
3
8 ∙ 27 = 2
∙3
Since 4 ∙ 25 = 4 ∙ 25 and 8 ∙ 27 =
Even and Odd Roots seem to follow the
same type of rule for multiplication.
3
3
P
=6
8 ∙ 27,
Multiplying and Simplifying
Radical Expressions
 Definition: Product Rule for Radicals
If
𝑛
𝑎 and
𝑛
𝑏 are real numbers,
𝑛
𝑛
𝑛
then
𝑎∙ 𝑏 ⇔ 𝑎∙𝑏.
(*index 𝑛 must be the same.)
Practice: Multiply and simplify.
3
5 ∙ 11
=
3
5 ∙ 11
=
3
55
a)
3
P
b)
𝑥+3∙ 𝑥−3
=
(𝑥 + 3) ∙ (𝑥 − 3)
=
𝑥2
−9
P
c)
=
=
7
3𝑥 2
7
7
3𝑥 2 ∙ 18𝑥 3
7
54𝑥 5
∙ 18𝑥 3
P
Multiplying and Simplifying
Radical Expressions
• Question?
Can the radicals below be multiplied directly?
3
6∙ 7
No, the index 𝑛 must be the same.
• Question?
Is 𝑥 2 − 9 the same as 𝑥 2 − 9?
Let 𝑥 = 5 to find out.(read p716)
Multiplying and Simplifying
Radical Expressions
• Simplify the Radical Expression by Factoring
and the Product Rule: ? 3 2 ∙ 40 ? 3 5 ∙ 16
3
3
3
4
∙
20
?
8 ∙ 10
?
80
3
3
80 = 8 ∙ 10
1) Factor radicand.
P
…make sure one factor of 80 is the
largest perfect 𝒄𝒖𝒃𝒆.
2) Apply Product Rule.
3) Simplify.
=
3
3
8 ∙ 10
3
= 2 ∙ 10
3
= 2 10
P
Multiplying and Simplifying
Radical Expressions
• Simplify the Radical Expression by Factoring
and the Product Rule:
5
64
5
5
64 = 32 ∙ 2
1) Factor radicand.
…make sure one factor is the
largest perfect 5𝑡ℎ power of 64.
2) Apply Product Rule.
3) Simplify.
=
5
5
32 ∙ 2
5
=2∙ 2
5
=2 2
P
Multiplying and Simplifying
Radical Expressions
• Simplify the Radical Expression by Factoring
and the Product Rule:
500𝑥𝑦 2
1) Factor radicand.
500𝑥𝑦 2 = 100 ∙ 5 ∙ 𝑥 ∙ 𝑦 2
=
100 ∙ 𝑦 2 ∙ 5 ∙ 𝑥
2) Apply Product Rule.
=
100𝑦 2 ∙ 5𝑥
3) Simplify.
= 10 𝑦 ∙ 5𝑥
p.696: Simplifying 𝑎2
= 10 𝑦 5𝑥
P
MATH 0322 Intermediate Algebra
Unit 2
Adding, Subtracting, Dividing
Radical Expressions
Section: 10.4
Adding, Subtracting, Dividing
Radical Expressions
• When adding or subtracting variable expressions,
like terms can be combined.
only ___
• In this section, like radical terms have:
index
1) the same _____
radicand
2) and the same ________.
• When combining like radical terms,
only the __________
coefficients are added or subtracted.
3
3
3
3
P
Example: 6 2𝑥 + 5 2𝑥 = 6 + 5 2𝑥 = 11 2𝑥
3
Example: 6 2𝑥 + 5 2𝑥 = Can’t simplify, index not the same
3
3
Example: 6 7𝑥 + 5 2𝑥 = Can’t simplify, radicand not the same
Adding, Subtracting, Dividing
Radical Expressions
Practice: Simplify by combining like radical terms.
3
3
3
3
P
a)
9 5+ 5 = 9+1
b)
6 𝑥 + 1 − 4 𝑥 + 1 + 7 𝑥 + 1 = (6 − 4 + 7) 𝑥 + 1
5 = 10 5
=9 𝑥+1
c)
=
7
3𝑥 2
7
3𝑥 2
7
18𝑥 3
7
3𝑥 2
7
3𝑥 2
+2
+4
= (1 + 4)
P
7
+ 4 3𝑥 2
7
+ 2 18𝑥 3
+2
7
18𝑥 3
=5
7
3𝑥 2
7
+ 2 18𝑥 3
P
Adding, Subtracting, Dividing
Radical Expressions
Practice: Simplify by combining like radical terms,
if possible.
a)
b)
5 12 − 6 27 = 5 4 ∙ 3 − 6 9 ∙ 3
factor
factor = 5 ∙ 2 3 − 6 ∙ 3 ∙ 3
= 10 3 − 18 3 = −8 3
3
5
7 2+9 2
Cannot be simplified.
P
P
Adding, Subtracting, Dividing
Radical Expressions
• Compare the following:
a)
36
4
36
4
= 9 =3
=
6
2
P
=3
b)
3
3
3
64
=
8
3
64
4
2
8
=
8 =2
=2
What do you notice?
Looks like a Rule for Dividing Radicals!
P
Adding, Subtracting, Dividing
Radical Expressions
 Definition: Quotient Rule for Radicals
If
𝑛
𝑎 and
then
𝑛
𝑎
𝑏
𝑛
𝑏 are real numbers and 𝑏 ≠ 0,
⇔
𝑛
𝑎
𝑛
𝑏
.
Read the “Great Question!” on page 727.
Adding, Subtracting, Dividing
Radical Expressions
Practice: Simplify using the Quotient Rule.
(Assume all variables represent positive real numbers.)
50
81
a)
Not square
Square
Factor
=
50
81
=
25∙2
81
=
5 2
9
P
b)
3
=
27𝑥 8
𝑦 12
3
27∙𝑥 8
3
𝑦 12
3
=
c)
3𝑥
𝑥2 𝑥 2
3
𝑦4
P
500𝑥 3
20𝑥 −1
=
500𝑥 3
20𝑥 −1
=
25𝑥𝑥33−(−1)
= 25𝑥 4
= 5𝑥 2
P
MATH 0322 Intermediate Algebra
Unit 2
Radical Equations
Section: 10.6
Radical Equations
• In this section, students will be asked to solve
Radical Equations by using:
Properties
of Exponents and Radicals,
P
Factoring
skills, Product Rule, Quotient Rule
P
and
P learned strategies for simplifying radical expressions.
• Radical Equation: an equation containing a
radicand of a radical expression.
variable in the ________
Examples:
𝑥 + 3 = 6,
𝑥 − 8 = 𝑥 − 2,
3
2𝑥 − 1 + 5 = 0
Radical Equations
• Think about what you already know about
…to…..hmmm,
get 𝑥 = 16,
the
value
of
𝑥
in
the
following
equation?
How
the
…so
this
how I
𝑎 = to
𝑏
…but
ifcan
Iissquare
I need
definition
of
can
solve
get
onlysquare
if 𝑎 = 4,…
𝑏 2 ….
the
right to
side,
𝑥 = 4
𝑏…
𝑥 𝑎=
I have
to=16
square
help
solve
for 𝑥?
Fan-tastic!
the
left side.
𝑥 2= 4 2
𝑥 = 16
𝑥 has to be 16,
because 16 = 4.
Radical Equations
• To solve a Radical Equation:
1) isolate the radical,
2) raise both sides to the 𝑛th power and simplify,
3) solve, then check solution(s).
(If equation contains more than one radical,
repeat Steps 1 & 2 until all radicals are eliminated.)
Radical Equations
Practice: Solve.
a)
Check:
3𝑥 + 4 2 = 8
3𝑥 + 4 = 64
3𝑥 = 60
𝑥 = 20
2
Step 1)
Step 2)
Step 3)
3(20)
20 + 4 = 8
60 + 4 = 8
P 64 = 8
P
Your turn.
3
b)
6𝑥 − 3 3= 3
6𝑥 − 3 = 27
6𝑥 = 30
𝑥=5
3
P
Check:
Radical Equations
Practice: Solve.
Check:
Step 1)
2𝑥 − 1 − 4 = 3
2𝑥 − 1 2= 7 2 Step 2)
2𝑥 − 1 = 49 Step 3)
2𝑥 = 50
𝑥 = 25
Your turn.
b)
6𝑥 + 3 + 15 = 24
6𝑥 + 3 2= 9 2
6𝑥 + 3 = 81
6𝑥 = 78
𝑥 = 13
2 20
25 − 1 − 4 = 3
a)
P
P
50 − 1 − 4 = 3
49 − 4 = 3
7−4=3
P
Check:
Radical Equations
Practice: Solve.
Check:
Step 1)
𝑥−1+7 =2
20
26 − 1 + 7 = 2
𝑥 − 1 2= −5 2 Step 2)
25 + 7 = 2
Step 3)
𝑥 − 1 = 25
5+7=2
𝑥 = 26
This is considered an “extraneous solution”.(See p745)
Your turn.
b)
7𝑥 + 8 + 15 = 9
Did you figure out when it
would not have a solution?
7𝑥 + 8 2 = −6 2
7𝑥 + 8 = 36
7𝑥 = 28
Check:
𝑥=4
“extraneous solution”
a)
O
O
O
Radical Equations
Practice: Solve.
a)
6𝑥 + 7 − 𝑥 = 2
6𝑥 + 7 2 = 𝑥 + 2 2
6𝑥 + 7 = 𝑥 2 + 4𝑥 + 4
7 = 𝑥 2 − 2𝑥 + 4
Step 1)
Step 2)
Step 3) Need Quadratic
Equation in Standard Form.
0 = 𝑥 2 − 2𝑥 − 3 Factor
0 = (𝑥 − 3)(𝑥 + 1)
𝑥−3 =0
𝑥=3
𝑥+1 =0
𝑥 = −1
P
P
Don’t forget to check for “extraneous solutions”.
Radical Equations
Practice: Solve.
b)
6𝑥 + 2 − 5𝑥 + 3 = 0
Step 1)
6𝑥 + 2 2 =
Step 2)
5𝑥 + 3
6𝑥 + 2 = 5𝑥 + 3
𝑥+2=3
P
2
Step 3)
𝑥=1
Don’t forget to check for “extraneous solutions”.
Radical Equations
Complete and
Practice HW 10.6.
MATH 0322 Intermediate Algebra
Unit 2
Complex Numbers
Section: 10.7
Complex Numbers
• In this section, students complete Chapter 10
by learning:
P
P
P
 what a Complex Number is,
 the components of a Complex Number,
 and how to operate with Complex Numbers.
• To prepare, we revisit −4 in Section 10.1.
Was −4 a real number? Yes or No
radicand must
Why? For even roots, the ________
be non-negative.
Complex Numbers
 Complex numbers: A number of the form
𝑎 + 𝑏𝑖
imaginary part
real part
 where 𝑎 and 𝑏 are real number coefficients,
 and 𝑖 is the imaginary unit.
P
P
 Definition: 𝑖 = −1
𝑖 2 = −1
𝑖 2 = −1
So what is
−4 ?
2
Product Rule
−4 = 4 ∙ −1 = 4 ∙ −1
= 2𝑖
imaginary
The square root of a negative number is ________.
 You will see this again in College Algebra.(learn it well)
Complex Numbers
Practice: Write as a multiple of 𝑖.
a)
−81
= 81 ∙ −1
= 81 ∙ −1
= 9𝑖
P
b)
−26
c)
−45
= 26 ∙ −1
= 2 ∙ 13 ∙ −1
= 45 ∙ −1
= 9 ∙ 5 ∙ −1
= 26 ∙ −1
= 9 ∙ 5 ∙ −1
= 26 𝑖
P
=3 5𝑖
P = 3𝑖
5
Your turn.
d)
−16
= 4𝑖
P
e)
−35
= 35 𝑖
P
f)
−28
=2 7𝑖
P = 2𝑖
7
Complex Numbers
 Operations with Complex numbers.
 To add or subtract, combine only like terms.
 To multiply, use Distributive Property with 𝑖 2 = −1.
 To divide, use a Conjugate to rationalize denominator.
 The conjugate of 2 − 5𝑖 is _______.
2 + 5𝑖
−4 − 9𝑖
 The conjugate of −4 + 9𝑖 is ________.
−3𝑖
 The conjugate of 3𝑖 is _____.
 All important in College Algebra-Chapter 5.
Complex Numbers
Practice: Perform the indicated operations.
Write the result in the form 𝑎 + 𝑏𝑖.
Combine like terms.
a)
2 − 𝑖 + (−6 + 3𝑖)
Real parts
b)
Imaginary parts
(2) + (−6) (−𝑖) + (3𝑖)
(−4)
(2𝑖)
+
−4 + 2𝑖
P
Your turn.
c) 9 − 5𝑖 + (−2 + 8𝑖)
7 + 3𝑖
P
−1 + 4𝑖 − (5 + 8𝑖)
−1 + 4𝑖 + (−5 − 8𝑖)
Real parts
Imaginary parts
(−1) + (−5) (4𝑖) + (−8𝑖)
(−6)
+
(−4𝑖)
−6 − 4𝑖
P
d)
−3 + 7𝑖 − (6 + 2𝑖)
−9 + 5𝑖
P
Complex Numbers
Practice: Perform the indicated operations.
Write the result in the form 𝑎 + 𝑏𝑖.
a) 5𝑖(−6 + 3𝑖)
b)
Distribute.
Distribute(FOIL).
−5 −4𝑖 +10𝑖 +8𝑖 2
−5 + 6𝑖 + 8𝑖 2
−5 + 6𝑖 + 8(−1)
−5 + 6𝑖 − 8
−13 + 6𝑖
−30𝑖 +15𝑖 2
−30𝑖 + 15(−1)
−15 − 30𝑖
P
Your turn.
c) 2𝑖(4 + 3𝑖)
−6 + 8𝑖
−1 + 2𝑖 (5 + 4𝑖)
P
P
d)
2 − 8𝑖 (−1 + 6𝑖)
46 + 20𝑖
P
Complex Numbers
Practice: Divide and simplify to form 𝑎 + 𝑏𝑖.
a)
2+𝑖 ∙
5−3𝑖5+3𝑖
Rationalize denominator
5+3𝑖
by multiplying conjugate.
=
10 + 6𝑖 + 5𝑖 + 3𝑖 2
25 + 15𝑖 − 15𝑖 − 9𝑖 2
=
10+11𝑖+3(−1)
25−9(−1)
=
10+11𝑖−3
25+9
=
=
7+11𝑖
34
7
11
+ 𝑖
34
34
P
Radical Equations
Complete and
Practice HW 10.7.
to prepare for Ch10 Exam.