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Transcript
Unit 4 Richardson Bellringer 9/23/14 1. Simplify: 64 = ±8 2. Simplify: 18π₯ 3 = 3π₯ 2π₯ Simplifying Radicals Review and Radicals as Exponents β’ A radical expression contains a root, which can be shown using the radical symbol, . β’ The root of a number x is a number that, when multiplied by itself a given number of times, equals x. β’ For Example: 2 3 π 4, 8, π₯ 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 2β2β2β3β3β3β5 3 2β2β2β3β3β3β5 3 2β3 5 3 6 5 β’ To add and/or subtract radicals you must first Simplify them, then combine like radicals. β’ Ex: 2 2 2 2 18 + 12 β 50 2 β 3 β 3 + 2 β 2 β 3 β 5 β 5 β 2 Simplifying 2 2 2 2 2 3 2+2 3β5 2 2 2 2 3β2 2 Radicals Adding and Subtracting Square Roots as Exponents Square Root Exponent 2 2 81 3β3β3β3 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 isnβt 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 β’ Can you take the square root of a negative number? β’ Ex: 2 β4 β what number times itself (π₯ 2 ) gives you a negative 4? β’ Can u take the cubed root of a negative number? β’ Ex: 3 β8 β what number times itself, and times (π₯ 3 ) itself again gives you a negative 8? β’ The imaginary unit i is used to represent the non-real value, 2 β1. β’ An imaginary number is any number of the form bi, where b is a real number, i = 2 β1, and b β 0. Exponent Rules Zero Exponent Property β’ A base raised to the power of 0 is equal to 1. β’ a0 = 1 Negative Exponent Property β’ A negative exponent of a number is equal to the reciprocal of the positive exponent of the number. βπ β’ π( π ) 1 π ( ) 1 π π = Exponent Rules Product of Powers Property β’ To multiply powers with the same base, add the exponents. β’ ππ β ππ = ππ+π Quotient of Powers Property β’ To divide powers with β’ the same base, subtract the exponents. π π ππ =π πβπ Exponent Rules Power of a Power Property Power of a Product Property β’ To raise one power to β’ To find the power of a another power, multiply the exponents. π π β’ (π ) = π πβπ product, distribute the exponent. π β’ (ππ) = π π β π π Exponent Rules Power of a Quotient Property β’ To find the power of a quotient, distribute the exponent. π π ππ β’( ) = π π π Bellringer 9/25/14 1. Simplify: 3 3 81 = 3 3 2. Simplify: (6 β π₯) β3 1 = 216π₯ 3 Imaginary Numbers and Exponents β’π= β’ π2 β’ π3 2 β1 2 2 2 3 2 2 4 2 = ( β1) = β1 = ( β1) = 2 β1 β ( β1) 2 2 2 β’ π 4 = ( β1) = ( β1) β ( β1) π5 2 = β1 β1 2 2 = β1 β β1 = 1 π 6 = β1 π8 = 1 = β1 2 π 7 = β1 β1 And so onβ¦ Roots and Radicals Review The Rules (Properties) Multiplication aο b ο½ Division a οb a ο½ b a b b may not be equal to 0. Roots and Radicals The Rules (Properties) Multiplication 3 aο b ο½ 3 3 Division a οb 3 3 a ο½ b 3 a b b may not be equal to 0. Roots and Radicals Review Examples: Multiplication 3 ο 3 ο½ 3ο3 ο½ 9 ο½3 Division 96 ο½ 6 96 6 ο½ 16 ο½ 4 Roots and Radicals Review Examples: Multiplication 3 Division 5 ο 16 ο½ 5 ο16 3 3 ο½ 3 80 3 270 ο½ 3 5 270 5 3 ο½ 8 ο10 ο½ 3 54 ο½ 3 27 ο 2 ο½ 8 ο 10 ο½ 3 27 ο 3 2 ο½ 2 10 ο½3 2 3 3 3 3 3 Intermediate Algebra MTH04 Roots and Radicals To add or subtract square roots or cube roots... β’ simplify each radical β’ 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 β’ 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. β’ Expressions containing imaginary numbers can also be simplified. β’ It is customary to put I in front of a radical if it is part of the solution. Simplifying with Complex Numbers Practice β’ Problem 1 β’ Problem 2 3 3 π+π π + π β π2 π + π β β1 πβπ =0 3 2 β8 + β8 (β2)(β2)(β2) + 3 β2 1 + 2 2 2 2 (2)(2)(2)(β1) 2 (β1) 2 β2 + 2 2 β β1 2 = β2 + 2π 2 Bellringer 9/26/14 1. Sub Rules Apply Practice With Sub β simplify, i, complex, exponent rules Bellringer 9/29/14 β’ Write all of these questions and your response 1. Is this your classroom? 2. Should you respect other peopleβs property and work space? 3. Should you alter Mrs. Richardsonβs 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? Letβs Review the real number system! β’ Rational numbers β’ Integers β’ Whole Numbers β’ Natural Numbers β’ Irrational Numbers More Examples of The Real Number System Now we have a new number! Complex Numbers Defined. β’ 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. β’ If the real part, a, is zero, then the complex number a +bi is just bi, so it is imaginary. β’ 0 + bi = bi , so it is imaginary β’ If the real part, b, is zero then the complex number a+bi is just a, so it is real. β’ a+ 0i =a , so it is real Examples β’ Name the real part of the complex number 9 + 16i? β’ What is the imaginary part of the complex numbers 23 - 6i? Check for understanding β’ Name the real part of the complex number 12+ 5i? β’ What is the imaginary part of the complex numbers 51 - 2i? β’ Name the real part of the complex number 16i? β’ What is the imaginary part of the complex numbers 23? β’ 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 β’ The square root of a negative number is an imaginary number. β’ You know that i = -1 β’ When n is some natural number (1,2,3,β¦), then -n = (-1)´n = i n Simply the following Negative Square Roots.. 1. -9 2. -16 3. -20 Letβs review the properties of exponentsβ¦. How could we make a list of i values? i0 = i = 1 i2 = i = 3 i4 = i = 5 i6 = Practice β’ Simply the following Negative Square Roots.. 1. -81 2. -144 3. -220 β’ Find the following i values.. 4. i 10 5. i 27 Bellringer 10/2/14 Simply the following Negative Square Roots: 1. β25 2. β18 3. 3 β24 How could we make a list of i values? i0 = i = 1 i2 = i = 3 i4 = i = 5 i6 = Note: β’A negative number raised to an even power will always be positive β’A negative number raised to an odd power will always be negative. How could we make a list of i values? i0 = 1 i = β1 = π 1 i 2 = π β π = β1 β β1 = β1 i = π 2 β π = β1 β β1 = βπ 3 2 )2 = β1 (π i = 4 2 2 2 (π i = ) β π = β1 5 2 )3 = β1 (π i = 6 3 = β1 β β1 = 1 2 β β1 = 1 β β1 = β1 = π = β1 β β1 β β1 = β1 Bellringer 10/3/14 β’ Turn in your Bellringers Bellringer 10/13/14 β’ Simplify the following: β’ π0 = 1 β’ π1 = β1 ππ π β’ π 2 = β1 β’ π 3 = βπ β’ π4 = 1 Review Review β Work on your own paper Review β Work on your own paper Bellringer 10/14/14 β’ Simplify the following: β’ 2 9 + 6π β’ π2 β’ π5 β’ 4π 2 β 8π 3 Review/practice Complex Numbers Bellringer 10/16/14 β’ Simplify the following: 1. πβππ‘ ππ π‘βπ π£πππ’π ππ π? 2. πβππ‘ ππ π‘βπ π£πππ’π ππ π 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 β’ Find the value of π 16 β’ Find the value of π 27 β’ Simplify β9 β’ Simplify β29 β’ What is π₯ exponentially? Bellringer 10/20/14 (7th) β’ Simplify the following: 1. πβππ‘ ππ π‘βπ π£πππ’π ππ π? 2. πβππ‘ ππ π‘βπ π£πππ’π ππ π 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 β’ Simplify: 1. β4π β 7π 2. β8 β β5 19 3. π Ex: ο 4i ο 7i ο½ ο28ο i ο½ 2 ο28ο ο1 ο½ 28 Remember i ο½ ο1 2 Ex# 2: ο8 ο ο5 ο½ i 8οi 5 ο½ Remember that ο1 ο½ i i ο 40 ο½ ο1ο 2 10 ο½ 2 ο2 10 Ex# 3: i 19 18 i ο½ i οi i οi ο½ ο¨i 18 19 ο© οi 9 2 ο¨i ο© ο i ο½ ο¨ο1ο© ο 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 ο c d and a b ο« c d are said to be conjugates of each other. Ex: 3 2i ο 5 and 3 2i ο« 5 Lets do an example: 8i Ex: 1 ο« 3i 8i 1 ο 3i ο 1 ο« 3i 1 ο 3i Rationalize using the conjugate Next 8i ο 24i 8i ο« 24 ο½ 1ο«9 10 2 4i ο« 12 5 Reduce the fraction Lets do another example 4ο«i Ex: 2i 4 ο« i i 4i ο« i ο ο½ 2 2i 2i i 2 Next 4i ο 1 4i ο« i ο½ 2 ο2 2i 2 Try these problems. 3 1. 2 ο 5i 3-i 2. 2-i 1. 2 ο« 5i 9 7ο«i 2. 5 Bellringer 10/21/14 1. What is π equivalent to? 2. What is π 2 equivalent to? 3. What is the Conjugate of 6 + 5π? Review Review: Simplify 1. π + 6π 7. 6 + π 3 β 2π 2. 4 + 6π + 3 4β6π 8. β1+π 3+4π 9. 2π 3. 5π β βπ 4. 5π β π β β2π 5. β6(4 β 6π) 6. β2 β π 4 + π Extra Review Review β Work on your own paper Review β Work on your own paper Review β Work on your own paper Review β Work on your own paper Review Review β Work on your own paper