Transcript Unit 4 - Bibb County Public School District

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
81
1
812
3∗3∗3∗3
Please put this in
your calculator.
What did you get?
=9
2
2
3*3
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?
2
−4 → what number times itself (𝑥 2 ) gives you a
negative 4?
• Ex:
• Can u take the cubed root of a negative number?
3
−8 → what number times itself, and times (𝑥 3 )
itself again gives you a negative 8?
• Ex:
• 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
3
270
5
 8 10
 3 54  3 27  2
 8  10
 3 27  3 2
 23 10
 33 2
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 (7th)
• 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 10/20/14
• Simplify:
1. −4𝑖 ∗ 7𝑖
2. −8 ∗ −5
19
3. 𝑖
Ex:  4i 7i 
28 i 
28 1
2
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
i  i i
9
18
2
i i  i   i
19
18
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
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
Extra Review
Review – Work on your own paper
Review – Work on your own paper
Review – Work on your own paper
Review – Work on your own paper
Review
Review – Work on your own paper