Unit 4
Richardson
Bellringer 9/23/14
1. Simplify
64 = ±8
2. Simplify
18𝑥 3 = 3𝑥 2𝑥
Simplifying Radicals Review
and Radicals as Exponents
• A radical expression contains
a root, which can be shown
using the radical symbol,
.
• The root of a number x is a
number that, when multiplied
by itself a given number of
times, equals x.
• For Example:
2
3
𝑛
4, 8, 𝑥
Simplifying
Radicals
Basic Review
Simplifying Radicals Steps
1. Use a factor tree to put the
number in terms of its
prime factors.
2. Group the same factor in
groups of the number on
the outside.
3. Merge those numbers into
1 and place on the outside.
4. Multiply the numbers
outside together and the
ones left on the inside
together.
3
1080
3
2∗2∗2∗3∗3∗3∗5
3
2∗2∗2∗3∗3∗3∗5
3
2∗3 5
3
6 5
• To add and/or subtract radicals
you must first Simplify them,
then combine like radicals.
• Ex:
2
2
2
2
18 + 12 − 50
2 ∗ 3 ∗ 3 + 2 ∗ 2 ∗ 3 − 5 ∗ 5 ∗ 2 Simplifying
2
2
2
2
2
3 2+2 3−5 2
2
2
2 3−2 2
Radicals
Adding and Subtracting
Square Roots as Exponents
Square Root
Exponent
2
2
81
3∗3∗3∗3
3*3
9
1
812
Please put this in
your calculator.
What did you get?
=9
Bellringer 9/24/14
Please get the calculator that has your seat
number on it, if there isn’t one please see me!
1. Simplify:
4
4
32 = 2 2
2. Rewrite as an exponent and solve on your
calculator:
5
1024 = 4
Exponent Rules and
Imaginary Numbers
- with multiplying and dividing square roots if we have time
Imaginary Numbers
• Can you take the square root of a negative number?
• Ex:
2
−4 → what number times itself (𝑥 2 ) gives you a
negative 4?
• Can u take the cubed root of a negative number?
• Ex:
3
−8 → what number times itself, and times (𝑥 3 )
itself again gives you a negative 8?
• The imaginary unit i is used to represent the non-real
value,
2
−1.
• An imaginary number is any number of the form bi,
where b is a real number, i =
2
−1, and b ≠ 0.
Exponent Rules
Zero Exponent Property
• A base raised to the
power of 0 is equal to 1.
• a0 = 1
Negative Exponent Property
• A negative exponent of
a number is equal to
the reciprocal of the
positive exponent of the
number.
−𝑚
• 𝑎( 𝑛 )
1
𝑚
( )
1
𝑎 𝑛
=
Exponent Rules
Product of Powers
Property
• To multiply powers
with the same base,
add the exponents.
• 𝑎𝑚 ∗ 𝑎𝑛 = 𝑎𝑚+𝑛
Quotient of Powers
Property
• To divide powers with
•
the same base, subtract
the exponents.
𝑚
𝑎
𝑎𝑛
=𝑎
𝑚−𝑛
Exponent Rules
Power of a Power
Property
Power of a Product
Property
• To raise one power to
• To find the power of a
another power, multiply
the exponents.
𝑚 𝑛
• (𝑎 ) = 𝑎
𝑚∗𝑛
product, distribute the
exponent.
𝑚
• (𝑎𝑏) =
𝑚
𝑎
∗
𝑚
𝑏
Exponent Rules
Power of a Quotient
Property
• To find the power of a
quotient, distribute the
exponent.
𝑎 𝑚 𝑎𝑚
•( ) = 𝑚
𝑏
𝑏
Bellringer 9/25/14
1. Simplify:
3
3
81 = 3 3
2. Simplify: (6 ∗ 𝑥)
−3
1
=
216𝑥 3
Imaginary Numbers and Exponents
•𝑖=
•
𝑖2
•
𝑖3
2
−1
2
2
2
3
2
2
4
2
= ( −1) = −1
= ( −1) =
2
−1 ∗ ( −1)
2
2
2
• 𝑖 4 = ( −1) = ( −1) ∗ ( −1)
𝑖5
2
= −1 −1
2
2
= −1 ∗ −1 = 1
𝑖 6 = −1
𝑖8 = 1
= −1
2
𝑖 7 = −1 −1
And so on…
Roots and Radicals Review
The Rules (Properties)
Multiplication
a b 
Division
a b
a

b
a
b
b may not be equal to 0.
Roots and Radicals
The Rules (Properties)
Multiplication
3
a b 
3
3
Division
a b
3
3
a

b
3
a
b
b may not be equal to 0.
Roots and Radicals Review
Examples:
Multiplication
3  3  33
 9 3
Division
96

6
96
6
 16  4
Roots and Radicals Review
Examples:
Multiplication
3
Division
5  16  5 16
3
3
 3 80
3
270

3
5
270
5
3
 8 10
 3 54  3 27  2
 8  10
 27  2
 2 10
3 2
3
3
3
3
3
3
3
Intermediate Algebra MTH04
Roots and Radicals
To add or subtract square roots or cube roots...
• simplify each radical
• add or subtract LIKE radicals by
adding their coefficients.
Two radicals are LIKE if they have the same expression under the
radical symbol.
Complex Numbers
Complex Numbers
• All complex numbers are of the form a + bi,
where a and b are real numbers and i is the
imaginary unit. The number a is the real part
and bi is the imaginary part.
• Expressions containing imaginary numbers
can also be simplified.
• It is customary to put I in front of a radical if
it is part of the solution.
Simplifying with Complex Numbers Practice
• Problem 1
• Problem 2
3
3
𝑖+𝑖
𝑖 + 𝑖 ∗ 𝑖2
𝑖 + 𝑖 ∗ −1
𝑖−𝑖
=0
3
2
−8 + −8
(−2)(−2)(−2) +
3
−2 1 + 2
2
2
2
(2)(2)(2)(−1)
2 (−1)
2
−2 + 2 2 ∗ −1
2
= −2 + 2𝑖 2
Bellringer 9/26/14
1.
Sub Rules Apply
Practice
With Sub – simplify, i, complex, exponent rules
Bellringer 9/29/14
• Write all of these questions and your response
1.
Is this your classroom?
2.
Should you respect other people’s property and work space?
3.
Should you alter Mrs. Richardson’s Calendar?
4.
How should you treat the class set of calculators?
Review Practice Answers
Discuss what to do when there is a substitute
Bellringer 9/30/14
*EQ- What are complex numbers? How can I distinguish
between the real and imaginary parts?
1. 1. How often should we staple our papers
together?
2. When should we turn in homework and where?
3. When and where should we turn in late work?
4. What are real numbers?
Let’s Review the real number system!
• Rational numbers
• Integers
• Whole Numbers
• Natural Numbers
• Irrational Numbers
More Examples of The Real Number System
Now we have a new number!
Complex Numbers Defined.
• Complex numbers are usually written in the form
a+bi, where a and b are real numbers and i is
defined as -1 . Because -1 does not exist in
the set of real numbers I is referred to as the
imaginary unit.
• If the real part, a, is zero, then the complex
number a +bi is just bi, so it is imaginary.
• 0 + bi = bi , so it is imaginary
• If the real part, b, is zero then the complex
number a+bi is just a, so it is real.
• a+ 0i =a , so it is real
Examples
• Name the real part of the complex number 9
+ 16i?
• What is the imaginary part of the complex
numbers 23 - 6i?
Check for understanding
• Name the real part of the complex number
12+ 5i?
• What is the imaginary part of the complex
numbers 51 - 2i?
• Name the real part of the complex number
16i?
• What is the imaginary part of the complex
numbers 23?
• Name the real part and the imaginary part
of each.
1. -4 - 3i
5
2.
20-11i
3.
18
2
4. 5 + i
3
5.
4-i
Bellringer
10/1/14
*EQ- How can I simplify the square root of a negative number?
For Questions 1 & 2, Name the real part and
the imaginary part of each.
1
1. -2 - i
2.
3
For Questions 3 & 4, Simplify each of the
following square roots.
9+ 4i
3.
12
4.
-1
Simply the following Square Roots..
1.
9
2.
25
3.
4.
24
32
How would you take the square root of a
negative number??
Simplifying the square roots with negative
numbers
• The square root of a negative number is an imaginary
number.
• You know that i =
-1
• When n is some natural number (1,2,3,…), then
-n = (-1)´n = i n
Simply the following Negative Square
Roots..
1.
-9
2.
-16
3.
-20
Let’s review the properties of exponents….
How could we make a list of i values?
i0 =
i =
1
i2 =
i =
3
i4 =
i =
5
i6 =
Practice
• Simply the following Negative Square Roots..
1.
-81
2.
-144
3.
-220
• Find the following i values..
4.
i
10
5.
i
27
Bellringer 10/2/14
Simply the following Negative Square Roots:
1. −25
2. −18
3. 3 −24
How could we make a list of i values?
i0 =
i =
1
i2 =
i =
3
i4 =
i =
5
i6 =
Note:
•A negative number raised
to an even power will
always be positive
•A negative number raised
to an odd power will
always be negative.
How could we make a list of i values?
i0 = 1
i = −1 = 𝑖
1
i 2 = 𝑖 ∗ 𝑖 = −1 ∗ −1 = −1
i = 𝑖 2 ∗ 𝑖 = −1 ∗ −1 = −𝑖
3
2 )2 = −1
(𝑖
i =
4
2
2 2
(𝑖
i = ) ∗ 𝑖 = −1
5
2 )3 = −1
(𝑖
i =
6
3
= −1 ∗ −1 = 1
2
∗ −1 = 1 ∗ −1 = −1 = 𝑖
= −1 ∗ −1 ∗ −1 = −1
Bellringer 10/3/14
• Turn in your Bellringers
Bellringer 10/13/14
• Simplify the following:
• 𝑖0 = 1
• 𝑖1 =
−1 𝑜𝑟 𝑖
• 𝑖 2 = −1
• 𝑖 3 = −𝑖
• 𝑖4 = 1
Review
Review – Work on your own paper
Review – Work on your own paper
Bellringer 10/14/14
• Simplify the following:
• 2 9 + 6𝑖
• 𝑖2
• 𝑖5
• 4𝑖 2 − 8𝑖 3
Review/practice Complex
Numbers
Bellringer 10/16/14
• Simplify the following:
1. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖?
2. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖 2 ?
3. Name the real and imaginary parts of the
following:
A. -2-I
B. 5+3i
C. 7i
D. 12
Bellringer 10/17/14
• Find the value of 𝑖 16
• Find the value of 𝑖 27
• Simplify
−9
• Simplify
−29
• What is
𝑥 exponentially?
Bellringer 10/20/14
• Simplify:
1. −4𝑖 ∗ 7𝑖
2. −8 ∗ −5
19
3. 𝑖
Ex:  4i  7i 
28 i 
2
28 1 
28
Remember i  1
2
Ex# 2:
8  5 
i 8i 5 
Remember that
1  i
i  40  1 2 10 
2
2 10
Ex# 3: i
19
18
i  i i
i i  i
18
19
 i
9
2
i   i  1  i
2 9
9
Answer: -i
Conjugate of Complex
Numbers
Conjugates
In order to simplify a fractional
complex number, use a conjugate.
What is a conjugate?
a b  c d and a b  c d
are said to be conjugates of
each other.
Ex: 3 2i  5 and 3 2i  5
Lets do an example:
8i
Ex:
1  3i
8i 1  3i

1  3i 1  3i
Rationalize using
the conjugate
Next
8i  24i
8i  24

19
10
2
4i  12
5
Reduce the fraction
Lets do another example
4i
Ex:
2i
4  i i 4i  i
 
2
2i
2i i
2
Next
4i  1
4i  i

2
2
2i
2
Try these problems.
3
1.
2  5i
3-i
2.
2-i
1.
2  5i
9
7i
2.
5
Bellringer 10/21/14
1. What is 𝑖 equivalent to?
2. What is 𝑖 2 equivalent to?
3. What is the Conjugate of 6 + 5𝑖?
Review: Simplify
1. 𝑖 + 6𝑖
7. 6 + 𝑖 3 − 2𝑖
2. 4 + 6𝑖 + 3
4−6𝑖
8.
−1+𝑖
3+4𝑖
9.
2𝑖
3. 5𝑖 ∗ −𝑖
4. 5𝑖 ∗ 𝑖 ∗ −2𝑖
5. −6(4 − 6𝑖)
6. −2 − 𝑖 4 + 𝑖
Bellringer 10/23/14
• What is 𝑖 equivalent to?
• Simplify 𝑖 12
• What is the conjugate of 3𝑖 − 2?
• Simplify
2𝑖
3𝑖−2
Review
Bellringer 10/24/14
1. Write “𝑖 =
−1” 20 times
2. Write “𝑖 2 = −1” 20 times
Review
Bellringer 10/28/14
• What is 𝑖 equivalent to?
• 𝑖2?
• Simplify 𝑖 21
• Simplify
16
• Essential Question: How are polynomial
operations related to operations in the complex
number system?
Bellringer 10/29/14
• Simplify 𝑖 13
• Simplify
9
• What is a Polynomial?
• What type of polynomial is 4𝑚3 − 6𝑚 + 2
• Essential Question: How are polynomial
operations related to operations in the complex
number system?
A polynomial is an algebraic
expression with one or more terms
A polynomial can have
constants (like 4 or -6),
variables (like x or y), and
exponents (like x2 or x100)
A polynomial can not have
negative exponents (like x-3) or
variables in the denominator (like 1/(x+2))
Adding and Subtracting
Polynomials
• Can only add or subtract LIKE TERMS
(terms having the same variables and
exponents)
• Add or subtract coefficients (leave
exponents the same)
EXAMPLES 1&2
Bellringer 10/30/14
• Simplify 𝑖 50
• Simplify
25
• What is a Coefficient?
• Add the polynomials:
•
5𝑥 2 + 7𝑥 3 − 4𝑥 4 + (2𝑥 2 − 2𝑥 4 + 8)
• Essential Question: How are polynomial operations
related to operations in the complex number system?
EXAMPLES 3&4
Multiplying Polynomials
• Multiply coefficients and add
exponents
• Terms do NOT need to be alike
(5)(x + 6)
5x  30
2
(x )(x
+ 6)
x  6x
3
2
2
(-2x)(x –
4x + 2)
2 x  8 x  4 x
3
2
When each polynomial
has 2 terms, distribute
each term in the first
polynomial to each term
in the second, then
combine like terms
(x + 5) (x + 3)
Bellringer 10/31/14
• Simplify 3𝑖 16
• Simplify
12
Subtract the polynomials:
•
5𝑥 2 + 7𝑥 3 − 4𝑥 4 − (2𝑥 2 − 2𝑥 4 + 8)
• Essential Question: How are polynomial
operations related to operations in the complex
number system?
Objective
The student will be able to:
multiply two polynomials using the
FOIL method, Box method and the
distributive property.
SOL: A.2b
Designed by Skip Tyler, Varina High School
There are three techniques you can use
for multiplying polynomials.
The best part about it is that they are all
the same! Huh? Whaddaya mean?
It’s all about how you write it…Here they
are!
1) Distributive Property
2) FOIL
3) Box Method
Sit back, relax (but make sure to write
this down), and I’ll show ya!
1) Multiply. (2x + 3)(5x + 8)
Using the distributive property,
multiply 2x(5x + 8) + 3(5x + 8).
10x2 + 16x + 15x + 24
Combine like terms.
10x2 + 31x + 24
A shortcut of the distributive
property is called the FOIL
method.
The FOIL method is ONLY used
when you multiply 2 binomials. It is
an acronym and tells you which
terms to multiply.
2) Use the FOIL method to
multiply the following binomials:
(y + 3)(y + 7).
(y + 3)(y + 7).
F tells you to multiply the FIRST terms of
each binomial.
y2
(y + 3)(y + 7).
O tells you to multiply the OUTER terms
of each binomial.
y2 + 7y
(y + 3)(y + 7).
I tells you to multiply the INNER
terms of each binomial.
y2 + 7y + 3y
(y + 3)(y + 7).
L tells you to multiply the LAST terms of
each binomial.
y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21
Remember, FOIL reminds you to
multiply the:
First terms
Outer terms
Inner terms
Last terms
The third method is the Box
Method. This method works for
every problem!
Here’s how you do it.
Multiply (3x – 5)(5x + 2)
3x
Draw a box. Write a polynomial on the
top and side of a box. It does not
matter which goes where.
This will be modeled in the next
problem along with FOIL.
5x
+2
-5
3) Multiply (3x - 5)(5x + 2)
First terms:
15x2
Outer terms:+6x
Inner terms: -25x
Last terms:
-10
Combine like
terms.
15x2
- 19x – 10
3x
5x
-5
15x2 -25x
+2 +6x
-10
You have 3 techniques. Pick the one you like the best!
4) Multiply (7p - 2)(3p - 4)
First terms:
21p2
Outer terms:-28p
Inner terms: -6p
Last terms:
+8
Combine like
terms.
21p2
– 34p + 8
7p
3p
-2
21p2 -6p
-4 -28p
+8
5) Multiply (2x - 5)(x - 5x + 4)
2
You cannot use FOIL because they
are not BOTH binomials. You
must use the distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
5) Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not
BOTH binomials. You must use the
distributive property or box method.
x2
-5x
+4
2x
2x3
-10x2
+8x
-5
-5x2 +25x
-20
Almost
done!
Go to
the next
slide!
5) Multiply (2x - 5)(x2 - 5x + 4)
Combine like terms!
x2
-5x
+4
2x
2x3
-10x2
+8x
-5
-5x2 +25x
-20
2x3 – 15x2 + 33x - 20
Bellringer 11/3/14
• Answer the Essential Question:
• How are polynomial operations related to operations
in the complex number system?
Bellringer 11/4/14
• Answer the Essential Question:
• How are rational exponents and roots of expressions
similar?
Bellringer 11/5/14
• No Bellringer :
• Clear your desk except for a pencil
Bellringer 11/6/14
• No Bellringer :
• Turn in your Review Packet
• Clear your desk except for a pencil
Bellringer 11/7/14
1. What is probability?
2. What is a subset?
3. What is the difference between a union and an
intersection?