DOCSLIB.ORG

EXPONENTS and RADICAL REVIEW Prep for Chapter 10 Exponents and Radicals Review I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
EXPONENTS and RADICAL REVIEW Prep for Chapter 10 Exponents and Radicals Review I EXPONENTS AND RADICAL REVIEW Prep for Chapter 10 Exponents and Radicals Review I. Inverse Operations Review OPERATION INVERSE Addition Subtraction Subtraction Addition Multiplication Division Division Multiplication Square Root Squaring Squaring Square Root Finding a root of a number is the inverse operation of raising a number to a power. (You can use other roots and powers besides 2). Exponents and Radicals Review II. Exponents The exponential form of a number is below. The exponent tells us how many times the base will be multiplied by itself. The base is the number being multiplied. A number produced by raising a base to an exponent is called a power. 27 and 33 are equivalent. Base Exponent 2 7 Exponents and Radicals Review Examples: Write in exponential form. A. 4 • 4 • 4 • 4• 4 Identify how many times 4 5 4 • 4 • 4 • 4 = 4 is a factor. B. (–6) • (–6) • (–6) Identify how many times –6 (–6) • (–6) • (–6) = (–6)3 is a factor. Reading Math Read (–6)3 as “–6 to the 3rd power" or "–6 cubed”. Exponents and Radicals Review III. Radical Expressions A. Terminology and Symbols This symbol is the radical or the radical sign index radical sign n a radicand The expression under the radical sign is the radicand. The index defines the root to be taken. Exponents and Radicals Review IV. Square Roots A square root of any positive number has two roots – one is positive and the other is negative. For example, the square root of 64 is both 8 and -8. When simplifying square roots (as opposed to solving them), we only use the positive, or principal, root. − The symbol The square root The above symbol represents the of a negative represents the negative root of a number is a non- positive or principal number. real number. root of a number. Ex: - 64 simplified Why? Ex: 64 simplified is -8 is 8 Exponents and Radicals Review To summarize, A negative outside the radical of a square root means you take the negative of the root. A negative inside the radical of a square root means it is a non-real number. Examples: 6 100 10 36 0.81 0.9 9 non-real # Exponents andRdicals Radicals Review V. Cube Roots 3 a A cubed root is asking “What base is multiplied three times to get the radicand?” In other words, “What times itself, times itself again, give you the radicand? Examples: 3 3 27 3 8 2 A cube root of any positive number is positive. A cube root of any negative number is negative. Exponents and Radicals Review th VI. n Roots An nth root of any number a is a number whose nth power is a. Examples: 34 81 4 81 3 24 16 4 16 2 5 2 32 5 32 2 Exponents and Radicals Review th n Roots Be careful with negatives inside the radical. Examples: Do you see a pattern? 5 1 1 If there is a negative under the radical and 4 16 Non-real number the index is positive, it is a non-real number. 6 If there is a negative 1 Non-real number under the radical and the index is negative, 3 27 3 the root will be negative. VII. PRIME FACTORIZATION Breaking down a number into its prime factorization will help you simplify radical expressions in 10-1. A. PRIME AND COMPOSITE NUMBERS A whole number, greater than 1, for which the only factors are 1 and itself is called a prime number. Examples: 2, 3, 5, 7, 11, 13, 17, 19 A whole number, greater than 1, that has more than two factors is called a composite number. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 20 0 and 1 are NEITHER prime nor composite. B. Finding the prime factorization of a number A whole number expressed as the product of prime numbers is called the prime factorization of the number. Ex. 1: Find the prime factorization of 90 Make a factor tree 90 What are some factors of 90? 9 10 Can we break down 9 and 10 further? Write factors for 3 3 5 2 each number that is composite. 90 = 2 ∙ 3 ∙ 3 ∙ 5 표푟 90 = 2 ∙ 32 ∙ 5 Circle the prime numbers. Notice how the numbers are in order from least to greatest. .
Load more

Recommended publications
  • The Five Fundamental Operations of Mathematics: Addition, Subtraction
    The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more.
    [Show full text]
  • 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]
  • Subtraction Strategy
    Subtraction Strategies Bley, N.S., & Thornton, C.A. (1989). Teaching mathematics to the learning disabled. Austin, TX: PRO-ED. These math strategies focus on subtraction. They were developed to assist students who are experiencing difficulty remembering subtraction facts, particularly those facts with teen minuends. They are beneficial to students who put heavy reliance on counting. Students can benefit from of the following strategies only if they have ability: to recognize when two or three numbers have been said; to count on (from any number, 4 to 9); to count back (from any number, 4 to 12). Students have to make themselves master related addition facts before using subtraction strategies. These strategies are not research based. Basic Sequence Count back 27 Count Backs: (10, 9, 8, 7, 6, 5, 4, 3, 2) - 1 (11, 10, 9, 8, 7, 6, 5, 4, 3) - 2 (12, 11, 10, 9, 8, 7, 6, 5,4) – 3 Example: 12-3 Students start with a train of 12 cubes • break off 3 cubes, then • count back one by one The teacher gets students to point where they touch • look at the greater number (12 in 12-3) • count back mentally • use the cube train to check Language emphasis: See –1 (take away 1), -2, -3? Start big and count back. Add to check After students can use a strategy accurately and efficiently to solve a group of unknown subtraction facts, they are provided with “add to check activities.” Add to check activities are additional activities to check for mastery. Example: Break a stick • Making "A" train of cubes • Breaking off "B" • Writing or telling the subtraction sentence A – B = C • Adding to check Putting the parts back together • A - B = C because B + C = A • Repeating Show with objects Introduce subtraction zero facts Students can use objects to illustrate number sentences involving zero 19 Zeros: n – 0 = n n – n =0 (n = any number, 0 to 9) Use a picture to help Familiar pictures from addition that can be used to help students with subtraction doubles 6 New Doubles: 8 - 4 10 - 5 12 - 6 14 - 7 16 - 8 18-9 Example 12 eggs, remove 6: 6 are left.
    [Show full text]
  • Number Sense and Numeration, Grades 4 to 6
    11047_nsn_vol2_add_sub_05.qxd 2/2/07 1:33 PM Page i Number Sense and Numeration, Grades 4 to 6 Volume 2 Addition and Subtraction A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 2006 11047_nsn_vol2_add_sub_05.qxd 2/2/07 1:33 PM Page 2 Every effort has been made in this publication to identify mathematics resources and tools (e.g., manipulatives) in generic terms. In cases where a particular product is used by teachers in schools across Ontario, that product is identified by its trade name, in the interests of clarity. Reference to particular products in no way implies an endorsement of those products by the Ministry of Education. 11047_nsn_vol2_add_sub_05.qxd 2/2/07 1:33 PM Page 1 Number Sense and Numeration, Grades 4 to 6 Volume 2 Addition and Subtraction A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 11047_nsn_vol2_add_sub_05.qxd 2/2/07 1:33 PM Page 4 11047_nsn_vol2_add_sub_05.qxd 2/2/07 1:33 PM Page 3 CONTENTS Introduction 5 Relating Mathematics Topics to the Big Ideas............................................................................. 6 The Mathematical Processes............................................................................................................. 6 Addressing the Needs of Junior Learners ..................................................................................... 8 Learning About Addition and Subtraction in the Junior Grades 11 Introduction.........................................................................................................................................
    [Show full text]
  • Solving for the Unknown: Logarithms and Quadratics ID1050– Quantitative & Qualitative Reasoning Logarithms
    Solving for the Unknown: Logarithms and Quadratics ID1050– Quantitative & Qualitative Reasoning Logarithms Logarithm functions (common log, natural log, and their (evaluating) inverses) represent exponents, so they are the ‘E’ in PEMDAS. PEMDAS (inverting) Recall that the common log has a base of 10 • Parentheses • Log(x) = y means 10y=x (10y is also known as the common anti-log) Exponents Multiplication • Recall that the natural log has a base of e Division y y Addition • Ln(x) = y means e =x (e is also know as the natural anti-log) Subtraction Function/Inverse Pairs Log() and 10x LN() and ex Common Logarithm and Basic Operations • 2-step inversion example: 2*log(x)=8 (evaluating) PEMDAS • Operations on x: Common log and multiplication by 2 (inverting) • Divide both sides by 2: 2*log(x)/2 = 8/2 or log(x) = 4 Parentheses • Take common anti-log of both sides: 10log(x) = 104 or x=10000 Exponents Multiplication • 3-step inversion example: 2·log(x)-5=-4 Division Addition • Operations on x: multiplication, common log, and subtraction Subtraction • Add 5 to both sides: 2·log(x)-5+5=-4+5 or 2·log(x)=1 Function/Inverse Pairs 2·log(x)/2=1/2 log(x)=0.5 • Divide both sides by 2: or Log() and 10x • Take common anti-log of both sides: 10log(x) = 100.5 or x=3.16 LN() and ex Common Anti-Logarithm and Basic Operations • 2-step inversion example: 5*10x=6 (evaluating) PEMDAS x • Operations on x: 10 and multiplication (inverting) • Divide both sides by 5: 5*10x/5 = 6/5 or 10x=1.2 Parentheses • Take common log of both sides: log(10x)=log(1.2) or x=0.0792
    [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]
  • Operation CLUE WORDS Remember, Read Each Question Carefully
    Operation CLUE WORDS Remember, read each question carefully. THINK about what the question is asking. Addition Subtraction add difference altogether fewer and gave away both take away in all how many more sum how much longer/ total shorter/smaller increase left less change Multiplication decrease each same twice Division product share equally in all (each) each double quotient every ©2012 Amber Polk-Adventures of a Third Grade Teacher Clue Word Cut and Paste Cut out the clue words. Sort and glue under the operation heading. Subtraction Multiplication Addition Division add each difference share equally both fewer in all twice product every gave away change quotient sum total left double less decrease increase ©2012 Amber Polk-Adventures of a Third Grade Teacher Clue Word Problems Underline the clue word in each word problem. Write the equation. Solve the problem. 1. Mark has 17 gum balls. He 5. Sara and her two friends bake gives 6 gumballs to Amber. a pan of 12 brownies. If the girls How many gumballs does Mark share the brownies equally, how have left? many will each girl have? 2. Lauren scored 6 points during 6. Erica ate 12 grapes. Riley ate the soccer game. Diana scored 3 more grapes than Erica. How twice as many points. How many grapes did both girls eat? many points did Diana score? 3. Kyle has 12 baseball cards. 7. Alice the cat is 12 inches long. Jason has 9 baseball cards. Turner the dog is 19 inches long. How many do they have How much longer is Turner? altogether? 4.
    [Show full text]
  • 1 Modular Arithmetic
    1 Modular Arithmetic Definition 1.1 (A divides B). Let A; B 2 Z. We say that A divides B (or A is a divisor of B), denoted AjB, if there is a number C 2 Z such that B = AC. Definition 1.2 (Prime number). Let P 2 N. We say that P is a prime number if P ≥ 2 and the only divisors of P are 1 and P . Definition 1.3 (Congruence modulo N). We denote by A mod N the remainder you get when you divide A by N. Note that A mod N 2 f0; 1; 2; ··· ;N − 1g. We say that A and B are congruent modulo N, denoted A ≡N B (or A ≡ B mod N), if A mod N = B mod N. Exercise 1.4. Show that A ≡N B if and only if Nj(A − B). Remark. The above characterization of A ≡N B can be taken as the definition of A ≡N B. Indeed, it is used a lot in proofs. Notation 1.5. We write gcd(A; B) to denote the greatest common divisor of A and B. Note that for any A, gcd(A; 1) = 1 and gcd(A; 0) = A. Definition 1.6 (Relatively prime). We say that A and B are relatively prime if gcd(A; B) = 1. Below, in Section 1.1, we first give the definitions of basic modular operations like addition, subtraction, multiplication, division and exponentiation. We also explore some of their properties. Later, in Section 1.2, we look at the computational complexity of these operations.
    [Show full text]