Name:
Block:
Unit 4: Part 2
Complex Numbers
Day1: Exponent Rules, Rational Exponents
Day2: Roots (without and with VARIABLES), a new number i,
and Solving Quadratics using the Square Root Method
Day3: Operations with Complex Numbers and Square Roots
Day4&5: Review
Day6: Quiz
Overview of the rest of this unit…
Where we’ve been –
Finding the vertex and graphing quadratics
given the equation in standard, vertex, and intercept form.
Where we’re going – What does it mean to SOLVE a quadratic equation?
Finding the zeros without the aid of a calculator, given a quadratic.
But before we can do that, we need…
An understanding of square roots and
complex and imaginary numbers
Day 1: Exponent Rules, Rational Exponents
We will:
I will:
A Handy Way to Remember What We’re About To Do…..
P
M
A
2x3x5
-3x2x7x4
Practice:
51  52 =
x2  5x6  x =
Does it really work?
26 =
Practice:
(25)3 =
(22)3 =
(x2)4 =
(23)2 =
Does it really work?
Use the property
(2 x 3)2 =
(2 x 3)2 =
(
)2
This DOES NOT work on addition.
(2 + 3)2=
COMMON MISTAKE: (x + 2)2
Practice:
(4x)3 =
(9x4y)2 =
Careful!
(-4z)2 =
–(4z)2 =
Does it really work? (Write the factors and cancel)
Use the property….
x6
=
x4
x6
=
x4
b8
=
b 14
b14
=
b8
3x7
=
6x4
2 y3
=
4 y11
x
 
 y
 4x 2

 5y
3



 7
 
 x
3
 a2 
 
 b 
x 7 =
x 2 y5
2
5
6x 0 =
5x 2 =
=
Watch this:
a.
x4
x 5
There are two ways to simplify this problem:
Put everything where it belongs first (move neg. exponent):
OR
b.
Use the Quotient of Powers rule:
x4
 x4 x5  x9
5
x
x4
 x 4( 5)  x 45  x 9
x 5
The first method usually creates the least amount of mistakes!!!! You try 
x6
x2
1.
2.
x 2
x 3
Key new points…
1.
2.
3.
You must have the same base before using the rules!
x 0 = 1 (anything to the zero power is 1)
1
x n = n means the reciprocal of xn .
x
Quick Questions: Which is correct?
Simplify
A
B
23  22
( x  4)2
160
(3x)2
 4 x 
 2
 5y 
2
 4x 
 2 
 5y 
 4 x 
 2
 5y 
2
3
 4x 
 2 
 5y 
45
( x  4)(x  4)
25
x 2  42
1
0
9x 2
1
9x 2

16 x 2
25 y 4
16 x 2
25 y 4

16 x 2
25 y 4
16 x 2
25 y 4
64 x3
125 y 3
64 x3
125 y 3
64 x3

125 y 3
64 x3
125 y 3

3
RATIONAL
EXPONENTS –
Same rules as
before, just fractions
1.
5 5
9
3
2
x x
4.
7.
1
4
3
3.
1
 23  2
5.  y 
 
6.
x x
4
3
5
1 1
11.    
4 4
13.
2
5
a b
1
2
1
8
3x3 y14
15 x5 y10
2
1
ab 3
4
 1 7
9.  x 2 
 
x2 y 5
8. 3
x y
x
x2
9
95
4
7
5
5
3
10.
 3a 
2.  2 
 6b 
12
14. x  x
5
6
3
12. (6 x 2 y 3 )0
 1
15.  2x 4 


3
Day 2: Simplifying Roots & Imaginary Numbers
We will:
I will:
So…….. about those rational exponents
Terminology & getting to know rational exponents
Exponential
Notation
x
x
x
x
x
1
Radical Notation
2
2
x1 or
1
3
1
2
3
3
4
4
3
4
am/n
(
3
(
4
x
You try:
Exponential
Notation
9
2
Radical Notation
3


x
x
x )2
or
x )3
or
49
3
x
3
1
n
am
5
x
2
3
, a0
Simplifying Radicals Using the Product Property (4.5):
Product Property of Radicals
2
2
( n a )m  n a m
a-m/n
5
4
4

84 
64
a b  a  b
*a > 0, b > 0
x9
What if the value is NOT a perfect square? This property helps us simplify radicals
Steps for simplifying square roots:
1. Make a list of perfect squares and keep it in sight!!!
1
4
9
16
25
36
49
64
81
100
121
144
169 …
2. Factor using the largest perfect square factor:
12 =
3. Simplify:
Be sure to use the largest perfect square!!!!
72
=
9
•
8
72
=
36
•
2
Simplify each expression. Remember … that means no decimals if it is not a perfect
square!
144
9
3
2
48
36
1
2
45
1
2
200
4

5
2
16
121
3
2
32
1
2
75
32

1
2
** You will eventually take a NO calculator section on quizzes & tests…so start memorizing and
recognizing your perfect squares!!!!
Simplify Radicals with Variables. Leave all answers in radical notation.
11a2b4c5
27x 5 y 8 z 3
144n6
What values of x do you think will make these equations true?
x2 = 81
x2 = 25
x2 = 49
x2 = -16
Complex Numbers: (4.6)
Not all quadratic equations have real number solutions. So we now get to learn about the
complex number system, which contains the imaginary unit i.
Imaginary Number:
i 
1
i2 

1

2
 1
The imaginary unit i can be used to write the square root of any negative number:
If r is a positive real number, then
r  i r
Example:
6  i 6
Simplify the Complex Numbers.
81
8
14
169

3

24
2
Solving Equations using Square Roots (4.5, 4.6)
Important Facts to remember when solving equations using square roots:




Isolate the exponent first!
Even Roots: 
Only use the imaginary # i when you are taking the SQUARE root of a NEGATIVE #
CHECK all solutions!!!
x2 = 49
2x2 = 50
3 (x – 5)2 = 27
(x + 3)2 = - 64
2x2 + 4 = 166
(x – 2) 2 = -75
1 2
x  2
4
2(x – 8)2 + 5 = - 45
(x + 1) 2 = 100
x2 = 32
Day 3: Complex Numbers and Roots –
Now that we have them, what can we do with them?
We will define operations on roots, complex numbers (& a few irrational numbers too!)
I will:
Warm up:
Review rational exponents and simplifying square roots – how are the following different?
Simplify them.
1
811/2
-1/2
1/2
81
81
-811/2
1
81-1/2
(-81)1/2
How are these last 2 different?
Operations on Roots
Addition and Subtraction
1. Need
2. Algorithm
34 3
5 2 3 2
12  7 3  4 5
5 20  4 5
3 2 8
18  50
Multiplication:
1. Need
2. Algorithm
2 8
7 6  2 
2 12  5 18 
34 3
3 3 6
5 2  4 10
 6  10
121  81
Division:
1. Need
2. Algorithm
20
5
18
9
What do we do if the values cannot be divided??? You
96
3
Rationalize Denominators
To rationalize a SQUARE ROOT – you need the denominator to be a PERFECT SQUARE (so
Multiply both numerator and denominator by the radical in the denominator)
1.
2.
3.
3

2
5
4 3
9 2
3

To rationalize a denominator with Two Terms – Multiply both numerator and denominator by a
conjugate. ( a  b and a  b are conjugates)
What happens when we multiply conjugates?
4.
(2  5)(2  5)
2
3 7
5.
3
52 3
Operations with Complex Numbers
Recall…..
Imaginary Unit _______
i1 = ____
i2 = ____
i3 = ____
i4 = ____
Complex Number _________________
Pure Imaginary Number _________
where a is known as _____________ and b is known as _______________
Complex Conjugates: (a + bi) & (a – bi)
What happens to the imaginary number?
(a + bi)(a – bi)
Operations with Complex Numbers (Add, Subtract, Multiply, Divide)
**Remember that i is a NUMBER that behaves like a VARIABLE
1. (4 + 2i) + (-3 + 5i)
2. (-7 – 6i) + (9 + 8i)
3. (15 + 2i) – (18 + i)
4. (21 + 7i) – (3 – 5i)
5. 6i – (8 + 9i) + (3 – 4i)
6. (9 + 4i) (5 – 3i)
7. (-8 – 2i) (11 + 6i)
8. (4i) (13 – 7i)
5
i
2  3i
10.
4i
5  4i
11.
2  3i
2  6i
12.
3  4i
9.
Discussion: How are operations with i similar/different from operations with square
roots?