DOCSLIB.ORG

Lesson 22: Multiplying and Dividing Expressions with Radicals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lesson 22: Multiplying and Dividing Expressions with Radicals NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Lesson 22: Multiplying and Dividing Expressions with Radicals Student Outcomes . Students multiply and divide expressions that contain radicals to simplify their answers. Students rationalize the denominator of a number expressed as a fraction. Lesson Notes Exercises 1–5 and the Discussion are meant to remind students of what they learned about roots in Grade 8 and Algebra I. In Grade 8, students learned the notation related to square roots and understood that the square root symbol automatically denotes the positive root (Grade 8, Module 7). In Algebra I, students used both the positive and negative roots of a number to find the location of the roots of a quadratic function. Because of the upcoming work with special triangles in this module, this lesson reviews what students learned about roots in Grade 8 Module 7 Lesson 4. For 1 √2 example, students need to understand that when they are writing the trigonometric ratios of right triangles. To √2 2 achieve this understanding, students must learn how to rationalize the denominator of numbers expressed as fractions. It is also important for students to get a sense of the value of a number. When a radical is in the denominator or is not simplified, it is more challenging to estimate its value, for example, √3750 compared to 25√6. For students who are struggling with the concepts of multiplying and dividing expressions with radicals, it may be necessary to divide the lesson so that multiplication is the focus one day and division the next. This lesson is a stepping- stone, as it moves students toward an understanding of how to rewrite expressions involving radical and rational exponents using the properties of exponents (N.RN.A.2), which are not mastered until Algebra II. The lesson focuses on simplifying expressions and solving equations that contain terms with roots. By the end of the lesson, students should understand that one reason the denominator of a number expressed as a fraction is rationalized 3 is to better estimate the value of the number. For example, students can more accurately estimate the value of √3 3√3 when written as or simply √3. Further, putting numbers in this form allows students to more easily recognize when 3 1 numbers can be combined. For example, if adding √3 and , students may not recognize that they can be combined √3 1 √3 √3 4√3 until is rewritten as . Then, the sum of √3 and is . As a teacher, it is easier to check answers when there is √3 3 3 3 an expected standard form such as a rationalized expression. Lesson 22: Multiplying and Dividing Expressions with Radicals 349 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Classwork Exercises 1–5 (8 minutes) The first three exercises review square roots that are perfect squares. The last two exercises require students to compare the value of two radical expressions and make a conjecture about their relationship. These last two exercises exemplify what is studied in this lesson. Students may need to be reminded that the square root symbol automatically denotes the positive root of the number. Scaffolding: Exercises 1–5 . Some students may need to review the perfect Simplify as much as possible. squares. A reproducible 1. √ퟏퟕퟐ = √ퟏퟕퟐ = ퟏퟕ sheet for squares of 1 30 numbers – is provided at the end of the lesson. ퟏퟎ √ퟓퟏퟎ = √ퟓퟐ × √ퟓퟐ × √ퟓퟐ × √ퟓퟐ × √ퟓퟐ 2. √ퟓ = . Consider doing a fluency = ퟓ × ퟓ × ퟓ × ퟓ × ퟓ activity that allows = ퟓퟓ students to learn their perfect squares up to 30. This may include choral 3. √ퟒ풙ퟒ = √ퟒ풙ퟒ = √ퟒ × √풙ퟐ × √풙ퟐ recitation. = ퟐ × 풙 × 풙 . English language learners = ퟐ|풙|ퟐ may benefit from choral practice with the word 4. Complete parts (a) through (c). radical. a. Compare the value of √ퟑퟔ to the value of √ퟗ × √ퟒ. The value of the two expressions is equal. The square root of ퟑퟔ is ퟔ, and the product of the square roots of ퟗ and ퟒ is also ퟔ. b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 풂 and 풃, √풂풃 = √풂 ∙ √풃. Explain. Answers will vary. Students should say that the statement √풂풃 = √풂 ∙ √풃 is valid because of the problem that was just completed: √ퟑퟔ = √ퟗ × √ퟒ = ퟔ. c. Does your conjecture hold true for 풂 = −ퟒ and 풃 = −ퟗ? No. The conjecture is not true when the numbers are negative because we cannot take the square root of a negative number. √(−ퟒ)(−ퟗ) = √ퟑퟔ = ퟔ, but we cannot calculate √−ퟒ × √−ퟗ in order to compare. 5. Complete parts (a) through (c). ퟏퟎퟎ √ퟏퟎퟎ a. Compare the value of √ to the value of . ퟐퟓ √ퟐퟓ ퟏퟎퟎ The value of the two expressions is equal. The fraction simplifies to ퟒ, and the square root of ퟒ is ퟐ. The ퟐퟓ ퟏퟎ square root of ퟏퟎퟎ divided by the square root of ퟐퟓ is equal to , which is equal to ퟐ. ퟓ Lesson 22: Multiplying and Dividing Expressions with Radicals 350 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 풂 and 풃, 풂 √풂 when 풃 ≠ ퟎ, √ = . Explain. 풃 √풃 풂 √풂 Answers will vary. Students should say that the statement √ = is valid because of the problem that was 풃 √풃 ퟏퟎퟎ √ퟏퟎퟎ just completed: √ = = ퟐ. ퟐퟓ √ퟐퟓ c. Does your conjecture hold true for 풂 = −ퟏퟎퟎ and 풃 = −ퟐퟓ? No. The conjecture is not true when the numbers are negative because we cannot take the square root of a −ퟏퟎퟎ √−ퟏퟎퟎ negative number. √ = √ퟒ = ퟐ, but we cannot calculate in order to compare. −ퟐퟓ √−ퟐퟓ Discussion (8 minutes) Debrief Exercises 1–5 by reminding students of the definition for square root, the facts given by the definition, and the rules associated with square roots for positive radicands. Whenever possible, elicit the facts and definitions from students based on their work in Exercises 1–5. Within this Discussion is the important distinction between a square root of a number and the square root of a number. A square root of a number may be negative; however, the square root of a number always refers to the principle square root or the positive root of the number. Definition of the square root: If 푥 ≥ 0, then √푥 is the nonnegative number 푝 so that 푝2 = 푥. This definition gives us four facts. The definition should not be confused with finding a square root of a number. For example, −2 is a square root of 4, but the square root of 4, that is, √4, is 2. Consider asking students to give an example of each fact using concrete numbers. Sample responses are included below each fact. Fact 1: √푎2 = 푎 if 푎 ≥ 0 √122 = 12, for any positive squared number in the radicand Fact 2: √푎2 = −푎 if 푎 < 0 This may require additional explanation because students see the answer as “negative 푎,” as opposed to the opposite of 푎. For this fact, it is assumed that 푎 is a negative number; therefore, −푎 is a positive number. It is similar to how students think about the absolute value of a number 푎: |푎| = 푎 if 푎 > 0, but |푎| = −푎 if 푎 < 0. Simply put, the minus sign is just telling students they need to take the opposite of the negative number 푎 to produce the desired result, that is, −(−푎) = 푎. √(−5)2 = 5, for any negative squared number in the radicand Fact 3: √푎2 = |푎| for all real numbers 푎 √132 = |13|, and √(−13)2 = | − 13| Fact 4: √푎2푛 = √(푎푛)2 = 푎푛 when 푎푛 is nonnegative 8 √716 = √(78)2 = 7 Lesson 22: Multiplying and Dividing Expressions with Radicals 351 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M2 GEOMETRY Consider asking students which of the first five exercises used Rule 1. When 푎 ≥ 0, 푏 ≥ 0, and 푐 ≥ 0, then the following rules can be applied to square roots: Rule 1: √푎푏 = √푎 ∙ √푏 A consequence of Rule 1 and the associative property gives us the following: √푎푏푐 = √푎(푏푐) = √푎 ⋅ √푏푐 = √푎 ∙ √푏 ∙ √푐 푎 √푎 Rule 2: √ = when 푏 ≠ 0 푏 √푏 We want to show that √푎푏 = √푎 ∙ √푏 for 푎 ≥ 0 and 푏 ≥ 0. To do so, we use the definition of square root. Consider allowing time for students to discuss with partners how they can prove Rule 1. Shown below are three proofs of Rule 1. Share one or all with the class. The proof of Rule 1: Let 푝 be the nonnegative number so that 푝2 = 푎, and let 푞 be the nonnegative number so that 푞2 = 푏. Then, √푎푏 = 푝푞 because 푝푞 is nonnegative (it is the product of two nonnegative numbers), and (푝푞)2 = 푝푞푝푞 = 푝2푞2 = 푎푏.
Load more

Recommended publications
  • Euler's Square Root Laws for Negative Numbers
    Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Complex Numbers Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 Euler's Square Root Laws for Negative Numbers Dave Ruch Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_complex Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Euler’sSquare Root Laws for Negative Numbers David Ruch December 17, 2019 1 Introduction We learn in elementary algebra that the square root product law pa pb = pab (1) · is valid for any positive real numbers a, b. For example, p2 p3 = p6. An important question · for the study of complex variables is this: will this product law be valid when a and b are complex numbers? The great Leonard Euler discussed some aspects of this question in his 1770 book Elements of Algebra, which was written as a textbook [Euler, 1770]. However, some of his statements drew criticism [Martinez, 2007], as we shall see in the next section. 2 Euler’sIntroduction to Imaginary Numbers In the following passage with excerpts from Sections 139—148of Chapter XIII, entitled Of Impossible or Imaginary Quantities, Euler meant the quantity a to be a positive number. 1111111111111111111111111111111111111111 The squares of numbers, negative as well as positive, are always positive. ...To extract the root of a negative number, a great diffi culty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of 4; we here require such as number as, when multiplied by itself, would produce 4; now, this number is neither +2 nor 2, because the square both of 2 and of 2 is +4, and not 4.
    [Show full text]
  • The Geometry of René Descartes
    BOOK FIRST The Geometry of René Descartes BOOK I PROBLEMS THE CONSTRUCTION OF WHICH REQUIRES ONLY STRAIGHT LINES AND CIRCLES ANY problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.1 Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract ther lines; or else, taking one line which I shall call unity in order to relate it as closely as possible to numbers,2 and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication) ; or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division) ; or, finally, to find one, two, or several mean proportionals between unity and some other line (which is the same as extracting the square root, cube root, etc., of the given line.3 And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater clearness. For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA ; then BE is the product of BD and BC.
    [Show full text]
  • A Quartically Convergent Square Root Algorithm: an Exercise in Forensic Paleo-Mathematics
    A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics David H Bailey, Lawrence Berkeley National Lab, USA DHB’s website: http://crd.lbl.gov/~dhbailey! Collaborator: Jonathan M. Borwein, University of Newcastle, Australia 1 A quartically convergent algorithm for Pi: Jon and Peter Borwein’s first “big” result In 1985, Jonathan and Peter Borwein published a “quartically convergent” algorithm for π. “Quartically convergent” means that each iteration approximately quadruples the number of correct digits (provided all iterations are performed with full precision): Set a0 = 6 - sqrt[2], and y0 = sqrt[2] - 1. Then iterate: 1 (1 y4)1/4 y = − − k k+1 1+(1 y4)1/4 − k a = a (1 + y )4 22k+3y (1 + y + y2 ) k+1 k k+1 − k+1 k+1 k+1 Then ak, converge quartically to 1/π. This algorithm, together with the Salamin-Brent scheme, has been employed in numerous computations of π. Both this and the Salamin-Brent scheme are based on the arithmetic-geometric mean and some ideas due to Gauss, but evidently he (nor anyone else until 1976) ever saw the connection to computation. Perhaps no one in the pre-computer age was accustomed to an “iterative” algorithm? Ref: J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity}, John Wiley, New York, 1987. 2 A quartically convergent algorithm for square roots I have found a quartically convergent algorithm for square roots in a little-known manuscript: · To compute the square root of q, let x0 be the initial approximation.
    [Show full text]
  • Unit 2. Powers, Roots and Logarithms
    English Maths 4th Year. European Section at Modesto Navarro Secondary School UNIT 2. POWERS, ROOTS AND LOGARITHMS. 1. POWERS. 1.1. DEFINITION. When you multiply two or more numbers, each number is called a factor of the product. When the same factor is repeated, you can use an exponent to simplify your writing. An exponent tells you how many times a number, called the base, is used as a factor. A power is a number that is expressed using exponents. In English: base ………………………………. Exponente ………………………… Other examples: . 52 = 5 al cuadrado = five to the second power or five squared . 53 = 5 al cubo = five to the third power or five cubed . 45 = 4 elevado a la quinta potencia = four (raised) to the fifth power . 1521 = fifteen to the twenty-first . 3322 = thirty-three to the power of twenty-two Exercise 1. Calculate: a) (–2)3 = f) 23 = b) (–3)3 = g) (–1)4 = c) (–5)4 = h) (–5)3 = d) (–10)3 = i) (–10)6 = 3 3 e) (7) = j) (–7) = Exercise: Calculate with the calculator: a) (–6)2 = b) 53 = c) (2)20 = d) (10)8 = e) (–6)12 = For more information, you can visit http://en.wikibooks.org/wiki/Basic_Algebra UNIT 2. Powers, roots and logarithms. 1 English Maths 4th Year. European Section at Modesto Navarro Secondary School 1.2. PROPERTIES OF POWERS. Here are the properties of powers. Pay attention to the last one (section vii, powers with negative exponent) because it is something new for you: i) Multiplication of powers with the same base: E.g.: ii) Division of powers with the same base : E.g.: E.g.: 35 : 34 = 31 = 3 iii) Power of a power: 2 E.g.
    [Show full text]
  • The Ordered Distribution of Natural Numbers on the Square Root Spiral
    The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution
    [Show full text]
  • Connecting Algebra and Geometry to Find Square and Higher Order Roots Iwan R
    Georgia Journal of Science Volume 72 No. 2 Scholarly Contributions from the Article 3 Membership and Others 2014 Connecting Algebra and Geometry to Find Square and Higher Order Roots Iwan R. Elstak [email protected] Sudhir Goel Follow this and additional works at: https://digitalcommons.gaacademy.org/gjs Part of the Mathematics Commons Recommended Citation Elstak, Iwan R. and Goel, Sudhir (2014) "Connecting Algebra and Geometry to Find Square and Higher Order Roots," Georgia Journal of Science, Vol. 72, No. 2, Article 3. Available at: https://digitalcommons.gaacademy.org/gjs/vol72/iss2/3 This Research Articles is brought to you for free and open access by Digital Commons @ the Georgia Academy of Science. It has been accepted for inclusion in Georgia Journal of Science by an authorized editor of Digital Commons @ the Georgia Academy of Science. Elstak and Goel: Connecting Algebra and Geometry to Find Roots 103 CONNECTING ALGEBRA AND GEOMETRY TO FIND SQUARE AND HIGHER ORDER ROOTS Iwan R. Elstak* and Sudhir Goel Valdosta State University, Valdosta, GA 31698 *Corresponding Author E-mail: [email protected] ABSTRACT Unfortunately, the repeat rate for all core curriculum courses including calculus-I and II, and also math education courses is too high. K-12 students are barely able to do estimation, an important part of Mathe- matics, whether it is adding fractions or finding roots, or for that matter, simple percent problems. In this paper we present geometric ways to reason about approximation of square roots and cube roots that are not accessible with simple routine techniques. We combine graphical methods, the use of a geometry software (sketch pad) and convergence of sequences to find higher order roots of positive real numbers and in the process, reason with recursion.
    [Show full text]
  • The Pythagorean Theorem
    6.2 The Pythagorean Theorem How are the lengths of the sides of a right STATES triangle related? STANDARDS MA.8.G.2.4 MA.8.A.6.4 Pythagoras was a Greek mathematician and philosopher who discovered one of the most famous rules in mathematics. In mathematics, a rule is called a theorem. So, the rule that Pythagoras discovered is called the Pythagorean Theorem. Pythagoras (c. 570 B.C.–c. 490 B.C.) 1 ACTIVITY: Discovering the Pythagorean Theorem Work with a partner. a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides (the legs) a and b. c2 c a a2 b. Label the length of the longest side (the hypotenuse) c. b b2 c . Draw squares along each of the three sides. Label the areas of the three squares a 2, b 2, and c 2. d. Cut out the three squares. Make eight copies of the right triangle and cut them out. a2 Arrange the fi gures to form 2 two identical larger squares. c b2 e. What does this tell you about the relationship among a 2, b 2, and c 2? 236 Chapter 6 Square Roots and the Pythagorean Theorem 2 ACTIVITY: Finding the Length of the Hypotenuse Work with a partner. Use the result of Activity 1 to fi nd the length of the hypotenuse of each right triangle. a. b. c 10 c 3 24 4 c. d. c 0.6 2 c 3 0.8 1 2 3 ACTIVITY: Finding the Length of a Leg Work with a partner.
    [Show full text]
  • The Square Root of I
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 2008 The Square Root of i Tiffany Lothrop University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Lothrop, Tiffany, "The Square Root of i" (2008). MAT Exam Expository Papers. 25. https://digitalcommons.unl.edu/mathmidexppap/25 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. The Square Root of i While Girolamo Cardano was working on solving cubic and quadratic equations in 1539, he encountered some formulas that involved square roots of negative numbers. In 1545 Cardano published Ars Magna, where he presents the first recorded calculations that involve complex numbers. Then in 1572, Rafael Bombelli published the first three parts of his Algebra. He is known as the inventor of complex numbers, because he identifies some rules for working with them. Bombelli also shows how complex numbers are very important and useful. From Bombelli’s list of rules for adding, subtracting and multiplying the complex numbers, he was able to analyze the cubic equations that Cardano was trying to solve in his paper Ars Magna. Bombelli was able to use his rules for operations with complex numbers to solve the cubic equations that produced an expression that contained a square root of a negative number.
    [Show full text]
  • Pythagoras' Theorem Information Sheet Finding the Length of the Hypotenuse
    Pythagoras’ Theorem In this activity you will use Pythagoras’ Theorem to solve real-life problems. Information sheet There is a formula relating the three sides of a right-angled triangle. It can be used to mark out right angles on sports pitches and buildings. If the length of the hypotenuse of a right-angled triangle is c and the lengths of the other sides are a and b: 2 2 2 c = a + b c 2 2 2 a and a = c – b 2 2 2 and b = c – a b To find the hypotenuse, add the squares of the other sides, then take the square root. To find a shorter side, subtract the squares of the other sides, then take the square root. Think about … How do you decide which sides to call a, b and c? How do you decide whether to add or subtract? Finding the length of the hypotenuse: example Find c. 12.4 cm 6.3 cm c How to do it …. Using Pythagoras gives c2 = 6.32 + 12.42 c 2 = 39.69 + 153.76 = 193.45 c = 193.45 = 13.9086… c = 13.9 cm (to 1 dp) Nuffield Free-Standing Mathematics Activity ‘Pythagoras Theorem’ Student sheets Copiable page 1 of 4 © Nuffield Foundation 2012 ● downloaded from www.fsmq.org Rollercoaster example C The sketch shows part of a rollercoaster ride between two points, A and B at the same horizontal level. drop The highest point C is 20 metres above AB, ramp 30 m the length of the ramp is 55 metres, and 55 m 20 m the length of the drop is 30 metres.
    [Show full text]
  • Chapter 9: Roots and Irrational Numbers
    Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic Equations G: Cube Roots Name: _____________________________________________________ Date: ____________________ Period: ______ Algebra Square Roots 9A Square roots, cube roots, and higher level roots are important mathematical tools because they are the inverse operations to the operations of squaring and cubing. In this unit we will study these operations, as well as numbers that come from using them. First, some basic review of what you’ve seen before. Exercise #1: Find the value of each of the following principal square roots. Write a reason for your answer in terms of a multiplication equation. (a) (b) (c) (d) (e) (f) It is generally agreed upon that all positive, real numbers have two square roots, a positive one and a negative one. We simply designate which one we want by either including a negative sign or leaving it off. Exercise #2: Give all square roots of each of the following numbers. (a) 4 (b) 36 (c) Exercise #3: Given the function , which of the following is the value of ? (1) 22 (3) 16 (2) 5 (4) 7 Square roots have an interesting property when it comes to multiplication. Lets discover that property. Exercise #4: Find the value of each of the following products. (a) (b) (c) (d) What you should notice in the last exercise is the following important property of square roots. MULTIPLICATON PROPERTY OF SQUARE ROOTS 1. LIKEWISE 2. One1. obviousa b use for a b this is to multiply two “unfriendlylikewise” square roots to get a nice2.
    [Show full text]
  • 3.5 Trigonometric Form of Complex Numbers
    3.5 Trigonometric Form of Complex Numbers • Plot complex numbers in a complex plane. • Determine the modulus and argument of complex numbers and write them in trigonometric form. • Multiply and divide two complex numbers in trigonometric form. • Use DeMoivre’s Theorem to ?ind powers of complex numbers. • Determine the nth roots of complex numbers. • What is the square root of i ? Are there more than one of them? 1 Review : What is i ? 2 3 4 i i i i Complex plane: i i i i Rectangular form of a complex number: a + bi z = 3+2i 1 z2 = 1- 4i Absolute value of a complex number: |a+bi | = √a2 + b2 Add two complex numbers: Multiply two complex numbers: 2 Trigonometric form of a complex number. z = a + bi becomes z = r(cos + isin ) r = |z| and the reference angle, ' is given by tan ' = |b/a| Note that it is up to you to make sure is in the correct quadrant. Example: Put these complex numbers in Trigonometric form. 4 - 4i -2 + 3i 3 Writing a complex number in standard form: Example: Write each of these numbers in a + bi form. √2 (cos 2π/3 + i sin 2π/3) 20 (cos 75º + i sin 75º) 4 Multiplying and dividing two complex numbers in trigonometric form: z z = r r (cos(ø +ø ) + i sin(ø +ø )) 1 2 1 2 1 2 1 2 z r 1 = 1 (cos(ø - ø ) + i sin(ø -ø )) z r 1 2 1 2 z1 = 3(cos 120º + i sin 120º) z2 = 12 (cos 45º + i sin 45º) 2 2 To multiply two complex numbers, you multiply the moduli and add the arguments.
    [Show full text]
  • History of Math Timeline Cards
    History of Math Timeline Cards Here’s how to make the cards: 1. Print the cards (color is nice, but not required). Each row has three columns. The first two columns are the front and back of the card. The last row is the source of the image, which is just FYI. 2. Cut out the first two columns of each row, then fold the card on the vertical column line so that it has a front and back. Just discard the last column of the card, unless you want to fold it onto the inside of the card. Note: I suggest removing some of the cards to tailor the topics to the knowledge level of your class. For example, my Geometry students may not yet know about trigonometry, so I would remove cards related to trigonometry for that class. 3. Put each card in a card protector (I bought these on Amazon). If you can see the date on the back of the card through the card protector, just slip a piece of cardboard (or these from Amazon) in the card protector. After sorting the cards, students can temporarily lift this cardboard to check the date. 4. You’re ready to go! Sources: 1. Katz, Victor J. A History of Mathematics: An Introduction. 2nd ed., Addison-Wesley Educational Publishers, Inc., 1998. 2. MacTutor History of Mathematics Archive Index to Chronology. School of Mathematics and Statistics, University of St. Andrews, Scotland, http://www-history.mcs.st-and.ac.uk/Chronology/index.html. Accessed 11 July 2018. 3. Richardson, William H.
    [Show full text]