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Unit 4 Richardson.

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1 Unit 4 Richardson

2 Bellringer 9/23/14 Simplify: 64 Simplify: 18 π‘₯ 3 =Β±8 =3π‘₯ 2π‘₯

3 Simplifying Radicals Review and Radicals as Exponents

4 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 4 , 3 8 , 𝑛 π‘₯ Simplifying Radicals Basic Review

5 Simplifying Radicals Steps
Use a factor tree to put the number in terms of its prime factors. Group the same factor in groups of the number on the outside. Merge those numbers into 1 and place on the outside. Multiply the numbers outside together and the ones left on the inside together. βˆ—2βˆ—2βˆ—3βˆ—3βˆ—3βˆ—5 2βˆ—

6 To add and/or subtract radicals you must first Simplify them, then combine like radicals.
Ex: βˆ’ 2 50 2 2βˆ—3βˆ— βˆ—2βˆ—3 βˆ’ 2 5βˆ—5βˆ—2 βˆ’5 2 2 2 2 3 βˆ’2 2 2 Simplifying Radicals Adding and Subtracting

7 Square Roots as Exponents
βˆ—3βˆ—3βˆ—3 3*3 9 Please put this in your calculator. What did you get? = 9

8 Bellringer 9/24/14 Please get the calculator that has your seat number on it, if there isn’t one please see me! Simplify: 4 32 Rewrite as an exponent and solve on your calculator: =2 4 2 =4

9 Exponent Rules and Imaginary Numbers
- with multiplying and dividing square roots if we have time

10 Imaginary Numbers Can you take the square root of a negative number?
Ex: 2 βˆ’4 β†’ what number times itself ( π‘₯ 2 ) gives you a negative 4? Can u take the cubed root of a negative number? Ex: 3 βˆ’8 β†’ what number times itself, and times ( π‘₯ 3 ) itself again gives you a negative 8? The imaginary unit i is used to represent the non-real value, 2 βˆ’1 . An imaginary number is any number of the form bi, where b is a real number, i = 2 βˆ’1 , and b β‰  0.

11 a0 = 1 π‘Ž ( βˆ’π‘š 𝑛 ) 1 = 1 π‘Ž ( π‘š 𝑛 ) Exponent Rules
Zero Exponent Property Negative Exponent Property A base raised to the power of 0 is equal to 1. a0 = 1 A negative exponent of a number is equal to the reciprocal of the positive exponent of the number. π‘Ž ( βˆ’π‘š 𝑛 ) 1 = 1 π‘Ž ( π‘š 𝑛 ) Examples:

12 π‘Ž π‘š π‘Ž 𝑛 = π‘Ž π‘šβˆ’π‘› π‘Ž π‘š βˆ— π‘Ž 𝑛 = π‘Ž π‘š+𝑛 Exponent Rules
Quotient of Powers Property Product of Powers Property To multiply powers with the same base, add the exponents. π‘Ž π‘š βˆ— π‘Ž 𝑛 = π‘Ž π‘š+𝑛 To divide powers with the same base, subtract the exponents. π‘Ž π‘š π‘Ž 𝑛 = π‘Ž π‘šβˆ’π‘› Examples:

13 ( π‘Ž π‘š ) 𝑛 = π‘Ž π‘šβˆ—π‘› (π‘Žπ‘) π‘š = π‘Ž π‘š βˆ— 𝑏 π‘š Exponent Rules
Power of a Power Property Power of a Product Property To raise one power to another power, multiply the exponents. ( π‘Ž π‘š ) 𝑛 = π‘Ž π‘šβˆ—π‘› To find the power of a product, distribute the exponent. (π‘Žπ‘) π‘š = π‘Ž π‘š βˆ— 𝑏 π‘š Examples:

14 ( π‘Ž 𝑏 ) π‘š = π‘Ž π‘š 𝑏 π‘š Exponent Rules
Power of a Quotient Property To find the power of a quotient, distribute the exponent. ( π‘Ž 𝑏 ) π‘š = π‘Ž π‘š 𝑏 π‘š Examples:

15 Bellringer 9/25/14 Simplify: 3 81 =3 3 3 Simplify: (6βˆ—π‘₯) βˆ’3
= π‘₯ 3

16 Imaginary Numbers and Exponents
𝑖= 2 βˆ’1 𝑖 2 = ( 2 βˆ’1 ) 2 =βˆ’1 𝑖 3 = ( 2 βˆ’1 ) 3 = 2 βˆ’1 βˆ—( 2 βˆ’1 ) 2 =βˆ’1 2 βˆ’1 𝑖 4 = ( 2 βˆ’1 ) 4 = ( 2 βˆ’1 ) 2 βˆ—( 2 βˆ’1 ) 2 =βˆ’1βˆ—βˆ’1=1 Patterns!! 𝑖 5 = 2 βˆ’ 𝑖 6 = βˆ’1 𝑖 7 =βˆ’1 2 βˆ’ 𝑖 8 = 1 And so on…

17 Roots and Radicals Review
The Rules (Properties) Multiplication Division b may not be equal to 0.

18 Roots and Radicals b may not be equal to 0. The Rules (Properties)
Multiplication Division b may not be equal to 0.

19 Roots and Radicals Review
Examples: Multiplication Division

20 Roots and Radicals Review Examples:
Multiplication Division

21 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.

22 Complex Numbers

23 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.

24 Simplifying with Complex Numbers Practice
Problem 1 𝑖+ 𝑖 3 𝑖+ π‘–βˆ— 𝑖 2 𝑖+ π‘–βˆ— βˆ’1 π‘–βˆ’π‘– =0 Problem 2 3 βˆ’8 + 2 βˆ’8 3 (βˆ’2)(βˆ’2)(βˆ’2) + 2 (2)(2)(2)(βˆ’1) βˆ’ (βˆ’1) βˆ’ βˆ— 2 βˆ’1 =βˆ’2+2𝑖 2 2

25 Bellringer 9/26/14 Sub Rules Apply

26 Practice With Sub – simplify, i, complex, exponent rules

27 Bellringer 9/29/14 Is this your classroom?
Write all of these questions and your response Is this your classroom? Should you respect other people’s property and work space? Should you alter Mrs. Richardson’s Calendar? How should you treat the class set of calculators?

28 Review Practice Answers
Discuss what to do when there is a substitute

29 Bellringer 9/30/14. EQ- What are complex numbers
Bellringer 9/30/14 *EQ- What are complex numbers? How can I distinguish between the real and imaginary parts? 1. How often should we staple our papers together? When should we turn in homework and where? When and where should we turn in late work? 4. What are real numbers?

30 Let’s Review the real number system!
Rational numbers Integers Whole Numbers Natural Numbers Irrational Numbers

31 More Examples of The Real Number System

32 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 Because 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

33 Examples Name the real part of the complex number 9 + 16i?
What is the imaginary part of the complex numbers i?

34 Check for understanding
Name the real part of the complex number i? What is the imaginary part of the complex numbers i? Name the real part of the complex number 16i? What is the imaginary part of the complex numbers 23?

35 Name the real part and the imaginary part of each.
1. 2. 3. 4. 5.

36 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. For Questions 3 & 4, Simplify each of the following square roots.

37 Simply the following Square Roots..
How would you take the square root of a negative number??

38 Simplifying the square roots with negative numbers
The square root of a negative number is an imaginary number. You know that i = When n is some natural number (1,2,3,…), then

39 Simply the following Negative Square Roots..

40 Let’s review the properties of exponents….

41 How could we make a list of i values?

42 Practice 1. 2. 3. Find the following i values.. 4. 5.
Simply the following Negative Square Roots.. 1. 2. 3. Find the following i values..

43 Bellringer 10/2/14 Simply the following Negative Square Roots: 1. βˆ’25 2. βˆ’ βˆ’24

44 How could we make a list of i values?

45 A negative number raised to an even power will always be positive
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.

46 How could we make a list of i values?
1 βˆ’1 =𝑖 π‘–βˆ—π‘–= βˆ’1 βˆ— βˆ’1 =βˆ’1 𝑖 2 βˆ—π‘–= βˆ’1 βˆ— βˆ’1 =βˆ’π‘– (𝑖 2 ) 2 = βˆ’1 2 =βˆ’1βˆ—βˆ’1=1 (𝑖 2 ) 2 βˆ—π‘–= βˆ’1 2 βˆ— βˆ’1 =1βˆ— βˆ’1 = βˆ’1 =𝑖 (𝑖 2 ) 3 = βˆ’1 3 =βˆ’1βˆ—βˆ’1βˆ—βˆ’1=βˆ’1

47 Bellringer 10/3/14 Turn in your Bellringers

48 Bellringer 10/13/14 Simplify the following: 𝑖 0 = 𝑖 1 = 1 𝑖 2 = 𝑖 3 =
𝑖 4 = 1 βˆ’1 π‘œπ‘Ÿ 𝑖 βˆ’1 βˆ’π‘– 1

49 Review

50 Review – Work on your own paper

51 Review – Work on your own paper

52 Bellringer 10/14/14 Simplify the following: 2 9 +6𝑖 𝑖 2 𝑖 5
𝑖 𝑖 2 𝑖 5 4 𝑖 2 βˆ’8 𝑖 3

53 Review/practice Complex Numbers

54 Bellringer 10/16/14 Simplify the following: π‘Šβ„Žπ‘Žπ‘‘ 𝑖𝑠 π‘‘β„Žπ‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝑖?
π‘Šβ„Žπ‘Žπ‘‘ 𝑖𝑠 π‘‘β„Žπ‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝑖 2 ? Name the real and imaginary parts of the following: -2-I 5+3i 7i 12

55 Bellringer 10/17/14 Find the value of 𝑖 16 Find the value of 𝑖 27
Simplify βˆ’9 Simplify βˆ’29 What is π‘₯ exponentially?

56 Bellringer 10/20/14 (7th) Simplify the following:
π‘Šβ„Žπ‘Žπ‘‘ 𝑖𝑠 π‘‘β„Žπ‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝑖? π‘Šβ„Žπ‘Žπ‘‘ 𝑖𝑠 π‘‘β„Žπ‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝑖 2 ? Name the real and imaginary parts of the following: -2-I 5+3i 7i 12

57 Bellringer 10/20/14 Simplify: βˆ’4π‘–βˆ—7𝑖 βˆ’8 βˆ— βˆ’5 𝑖 19

58 Remember 28

59

60 Answer: -i

61 Conjugate of Complex Numbers

62 Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

63 are said to be conjugates of each other.

64 Lets do an example: Rationalize using the conjugate Next

65 Reduce the fraction

66 Lets do another example
Next

67 Try these problems.

68

69 Bellringer 10/21/14 What is 𝑖 equivalent to?
What is the Conjugate of 6+5𝑖?

70 Review

71 𝑖+6𝑖 6+𝑖 3βˆ’2𝑖 4+6𝑖+3 4βˆ’6𝑖 βˆ’1+𝑖 5π‘–βˆ—βˆ’π‘– 3+4𝑖 2𝑖 5π‘–βˆ—π‘–βˆ—βˆ’2𝑖 βˆ’6(4βˆ’6𝑖)
Review: Simplify 𝑖+6𝑖 6+𝑖 3βˆ’2𝑖 4+6𝑖+3 4βˆ’6𝑖 βˆ’1+𝑖 5π‘–βˆ—βˆ’π‘– 3+4𝑖 2𝑖 5π‘–βˆ—π‘–βˆ—βˆ’2𝑖 βˆ’6(4βˆ’6𝑖) βˆ’2βˆ’π‘– 4+𝑖

72 Extra Review

73 Review – Work on your own paper

74 Review – Work on your own paper

75 Review – Work on your own paper

76 Review – Work on your own paper

77 Review

78 Review – Work on your own paper


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