DOCSLIB.ORG

Rational Exponents and Radicals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rational Exponents and Radicals 26 ChapterP r rn Prerequisites Rational Exponentsand Radicals Raising a number to a power is reversedby finding the root of a number. we indicate roots by using rational exponentsor radicals. In this section we will review defini- tions and rules concerning rational exponentsand radicals. Roots Since2a : 16 and (-D4 : 16,both 2 and -2 are fourth roots of 16.The nth root of a number is defined in terms of the nth Dower. Definition:nth Roots lf n is a positive integer and an : b, then a is called an nth root of b. If a2 : b, thena is a squareroot ofD. lf a3 : b,thena is the cube root ofb. we also describeroots as evenor odd"depending on whetherthe positive integeris even or odd. For example,iffl is even (or odd) and a is an nth root ofb, then a is called an even (or odd) root of b. Every positive real number has two real even roots, a positive root and a negativeroot. For example,both 5 and -5 are square roots of 25 because52 :25 and (-5)2 : 25. Moreover,every real numberhas exactly onereal odd root. For example,because 23 : 8 and 3 is odd,2 is the only real cuberoot of 8. Because(-2)3 : -8 and 3 is odd, -2 is the only real cube root of -8. Finding an nth root is the reverseof finding an nth power, so we use the notation at/n for the nth root of a. For example, since the positive squareroot of 25 is 5, we write25t/2 : 5. Definition:Exponent 1 /n If z is a positive even integer and a is positive, then at/n denotes the positive real zth root ofa and is called the principal zth root ofa" If n is a positive odd integer and a is any real number, then arln denotesthe real nth root ofa. If n is a positive integer,then 0lln : 0. fua.*V/e I rvaluatingexpressions involving exponent 1/n Evaluate each expression. a. 4tl2 b. gr/3 c. (-g;t/: d. (+yrlz Solution a. The expression4ll2 representsthe positivereal squareroot of 4. So 4tl2 : 2. b. 8t/3 : 2 c. (_g)t/3: _2 P.3r ilffi RationalExoonents and Radicals 27 d. Since the definition of nth root does not include an even root of a negative num- ber, (_471/zhas not yet beendefined. Even roots of negativenumbers do exist in the complex number system,which we define in Section P.4.So an even root of a negative number is not a real number. TrVThs.Evaluate. a. 91/2 b. l6tl4 Rational Exponents We havedefined otl" asthe nthroot of a. We extendthis definitionto e'ln,which is defined as the mth power of the nth root of a. A rational exponent indicates both a root and a power. Definition: If m andr arepositive integers, then RationalExponents e^ln : @rlr)^ providedthat all' is a realnumber. Note that a'tn is not real when a is negative and n is even. According to the defini- tion of rationalexponents, expressions such as (_251-zlz, (*+31t1+,and(- l)21'ate not defined becauseeach ofthem involves an even root ofa nesative number. Note thatsomeauthorsdefinea'l' onlyfor mfninlowestterms.In-thatcasethefourth power of the square root of 3 could not be written as 3a/2.This author prefers the more general definition given above. The root and the power indicated in a rational exponent can be evaluated in n In l' either order.That is, (ot I )' : (onlt providedat is real.For example, g2l3:(Btlrz:22:4 or g2l3 : (92)tl3 : 64t13: 4. A negative rational exponent indicates reciprocal just as a negative integral exponentdoes. So 1 111 ' - - o Jta 6-t- (gt/3)2 22 4' Evaluate8-213 mefrtally as follows: The cube root of 8 is 2,2 squaredis 4, and the reciprocal of 4 is j. The three operations indicated by a negative rational exponent can be performed in any order, but the simplest procedure for mental evaluationsis summarized as follows. WWWWWWWWW Evaluatinsd-mtn To evaluatea-'ln mentally, 1. find the nth root of a, 2. raiseit to the m powel 3. find the reciprocal. 28 Chapter P I rffi Prerequisites Rational exponentscan be reduced to lowest terms. For example, we can evalu- arc 2612by first reducing the exponent: 2612:23:B Exponents can be reduced only on expressionsthat are real numbers. For example, - t I G g'1' + ( l) because(- 9'1t is not a real number,while ( - I ) is a real number. - ru Your graphing calculatorwill probably evaluate\- t7z1zas 1, becauseit is not using our definition. Moreover, some calculators will not evaluatean expression with a negative base such as (-8;z/:, but will evaluate the equivalent expression ((-8)')1/3.To useyour calculatoreffectively, you must get to know it well. tr Arc*V/e' I Evatuatingexpressions with rational exponents Evaluate each expression. a. (Z1zlz b. 27-213 c. 1006/a Solution a. Mentally,the cuberoot of -8 is -2 and the squareof -2 is 4. In symbols: (-21'1t: ((-8)tn)t : (-D2 : 4 -8) b. Mentally,the cube rootof 27 is 3, the squareof 3 is 9, and the reciprocalof 9 is j. { ) In symbols: ( -?/3)rt*rtr, 4 .>t-zl3- I 1-1 L' - - lBE^(6/4) g lEBE (27tlty 32 c. 1006/a: 100312: 103: 1000 wFigure P.20 FE fne expressionsare evaluatedwith a graphing calculator in Fig. P.20. TrV 77ar. Evaluate.a. 9312b. 16-sl4 Rulesfor Rational Exponents The rules for integral exponents from Section P.2 also hold for rational exponents. Rulesfor The followingrules are valid for all real nunbersa andb andrational numbers RationalExponents r ands, providedthat all indicatedpowers are real and no denominatoris zero. : l. a'e" o'*' 2. { = o'-' 3. (ar)s * qrs a- / o\' a' - ( o\-' / t\' o-' 4. (ab),: a,b, t.\;) : u. : ,,. =bt (;/ (;/ b_, a' " When variable expressionsinvolve even roots, we must be careful with signs. For example, (xz\tlz : * ," not correct for all values of x, because (-51z1ttz : 25t12: 5. Howevequsing absolutevalue we can write G1112: lxl for everyreal number x P.3r ffiw RatronalExoonents and Radicals 29 When finding an even root of an expressioninvolving variables, remember that if n is even,a'/' islhe positive nthroot of rz. footn//a I usingabsotute value with rational exponents Simplify each expression, using absolute value when necessary.Assume that the variables can representany real numbers. a. (64a\U6 b. Qett/t c. @8Stl+ d. (yt')tlo Sotution a. For any nonnegativereal number a,wehave (64aa1rle:2a. If a is negative, (64aa1tleis positive and2a is negative.So we write (6+a\tla : 2ol : 2lol for everyreal number a. b. Foranynonnegativex,wehave(x\tlz: xel3- x3.lf xisnegative,(*e)113 and 13 are both negative. So we have Qslrlz : *z for every real numberx. c. For nonnegativec, we have @t\t1+ : a2. Since (o')'lo anda2 areboth positiveif a is negative,no absolutevalue sign is needed.So (a\tl4 : a2 for every real numbera. d. Fornonnegativey,wehave(yt')tlo: rz.rcyisnegative, (y'\'lo ispositivebut y3 is negative.So (yt\tlo : lyt l for every real numbery. 4rV Titl. Let w representany real number. Simplify (r\tb When simplifying expressionswe often assumethat the variablesrepresent pos- itive real numbers so that we do not have to be concerned about undefined expres- sions or absolute value. In the following example we make that assumption as we use the rules of exponentsto simplify expressionsinvolving rational exponents. fuonF/e I Simplifyingexpressions with rational exponents Use the rules of exponents to simplify each expression.Assume that the variables representpositive real numbers.Write answerswithout negative exponents. ( aztz6ztzlt a. xzl3x4l3 b. @aytlzlrl+ c. tr, \4-./ Sotution L. x2l3x4l3 : x6/3 Procluctrule _ --2 -L Simplitythe exponcnt. b. (xayllzltl+: G\t14(yll\t14 Power of a productrule : *!'l' Powero1'a power lule ChapterP rrr Prerequisites ( a3l2b2/3\3 6t/z1z162/t1z : --@- Powerofaquotientrule "' l-7- ) oe1262 Powerofa powerrule u^d lo -;)?\ a-5lLbz QuotientruleV- 6 = D- 7t) Definition of negativeexponents a't' TrV 1ht. Simpliff(ar/zor/z7rz. Radical Notation The exponentlf n andtheradical sign V- are both usedto indicatethe nth root. Definition:Radical ',1 | The numbera is calledthe radicand andn is the index of the radical.Expressions suchas \/4, {-u, and{/i do not representreal numbersbecause each is an evenroot of a negative.number.. ho*Vle p Evaluatingradicals Evaluateeach expression and cheokwith a calculator. o116 ,. J1q9b. tf looo t'V81 $olution a. Thesymbol \6 indicatesthe positive squaxe root of 49. So \6 : 49112: 7. Writing\/49 : +7 is incorrect. b. t'f 1000= (-1000)1/3: -10 checkthat(-ro;t: -tooo. olrc (rc\'ro 2 '' : :1 checkthat(3I = V8r \sr/ *f r FigureP.21 Theseexpressions are evaluated with a calculatorin Fig. P.21. TrV Tltt. Evaluate. Vioo b. */-n ". Sinceall' I t- :, Y^nf a, expressionsinvolving rational exponentscan be written with radicals. P.3r rr RationalExponents and Radicals 31 Rule:Converting atln to RadicalNotation fua*T/a @ writing rational exponents as radicals Write eachexpression in radicalnotation. Assume that all variablesrepresent posi- tive realnumbers. Simplify the radicandif possible. t Z2l3 b. (3g3la c. 2(x2 + 3)-112 Sotution a. 22/3: t/F : t/+ b. (lx1z/+: {-Arf : tfr*t c. 2(* + 31-t1z (x" + 31'r" t/f + I TrV \hl. Wirte 5213in radicalnotation. r The Product and Quotient Rules for Radicals Using rational exponentswe can write - (ab)r/' o1/n61/n andc)''':# Theseequations say that the nth root ofa product (or quotient)is the product(or quotient) of the zth roots. Using radical notation these rules are written as follows. Rule:Product and Quotient An expressionthat is the squareof a term that is free of radicalsis called a perfect square..Forexample, gx6 is a perfectsquaxe because ,*e : (3x3)2.Like- wise,27yr2is a perfect cube.
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]
  • The Nth Root of a Number Page 1
    Lecture Notes The nth Root of a Number page 1 Caution! Currently, square roots are dened differently in different textbooks. In conversations about mathematics, approach the concept with caution and rst make sure that everyone in the conversation uses the same denition of square roots. Denition: Let A be a non-negative number. Then the square root of A (notation: pA) is the non-negative number that, if squared, the result is A. If A is negative, then pA is undened. For example, consider p25. The square-root of 25 is a number, that, when we square, the result is 25. There are two such numbers, 5 and 5. The square root is dened to be the non-negative such number, and so p25 is 5. On the other hand, p 25 is undened, because there is no real number whose square, is negative. Example 1. Evaluate each of the given numerical expressions. a) p49 b) p49 c) p 49 d) p 49 Solution: a) p49 = 7 c) p 49 = unde ned b) p49 = 1 p49 = 1 7 = 7 d) p 49 = 1 p 49 = unde ned Square roots, when stretched over entire expressions, also function as grouping symbols. Example 2. Evaluate each of the following expressions. a) p25 p16 b) p25 16 c) p144 + 25 d) p144 + p25 Solution: a) p25 p16 = 5 4 = 1 c) p144 + 25 = p169 = 13 b) p25 16 = p9 = 3 d) p144 + p25 = 12 + 5 = 17 . Denition: Let A be any real number. Then the third root of A (notation: p3 A) is the number that, if raised to the third power, the result is A.
    [Show full text]
  • Math Symbol Tables
    APPENDIX A I Math symbol tables A.I Hebrew letters Type: Print: Type: Print: \aleph ~ \beth :J \daleth l \gimel J All symbols but \aleph need the amssymb package. 346 Appendix A A.2 Greek characters Type: Print: Type: Print: Type: Print: \alpha a \beta f3 \gamma 'Y \digamma F \delta b \epsilon E \varepsilon E \zeta ( \eta 'f/ \theta () \vartheta {) \iota ~ \kappa ,.. \varkappa x \lambda ,\ \mu /-l \nu v \xi ~ \pi 7r \varpi tv \rho p \varrho (} \sigma (J \varsigma <; \tau T \upsilon v \phi ¢ \varphi 'P \chi X \psi 'ljJ \omega w \digamma and \ varkappa require the amssymb package. Type: Print: Type: Print: \Gamma r \varGamma r \Delta L\ \varDelta L.\ \Theta e \varTheta e \Lambda A \varLambda A \Xi ~ \varXi ~ \Pi II \varPi II \Sigma 2: \varSigma E \Upsilon T \varUpsilon Y \Phi <I> \varPhi t[> \Psi III \varPsi 1ft \Omega n \varOmega D All symbols whose name begins with var need the amsmath package. Math symbol tables 347 A.3 Y1EX binary relations Type: Print: Type: Print: \in E \ni \leq < \geq >'" \11 « \gg \prec \succ \preceq \succeq \sim \cong \simeq \approx \equiv \doteq \subset c \supset \subseteq c \supseteq \sqsubseteq c \sqsupseteq \smile \frown \perp 1- \models F \mid I \parallel II \vdash f- \dashv -1 \propto <X \asymp \bowtie [Xl \sqsubset \sqsupset \Join The latter three symbols need the latexsym package. 348 Appendix A A.4 AMS binary relations Type: Print: Type: Print: \leqs1ant :::::; \geqs1ant ): \eqs1ant1ess :< \eqs1antgtr :::> \lesssim < \gtrsim > '" '" < > \lessapprox ~ \gtrapprox \approxeq ~ \lessdot <:: \gtrdot Y \111 «< \ggg »> \lessgtr S \gtr1ess Z < >- \lesseqgtr > \gtreq1ess < > < \gtreqq1ess \lesseqqgtr > < ~ \doteqdot --;- \eqcirc = .2..
    [Show full text]
  • Using XYZ Homework
    Using XYZ Homework Entering Answers When completing assignments in XYZ Homework, it is crucial that you input your answers correctly so that the system can determine their accuracy. There are two ways to enter answers, either by calculator-style math (ASCII math reference on the inside covers), or by using the MathQuill equation editor. Click here for MathQuill MathQuill allows students to enter answers as correctly displayed mathematics. In general, when you click in the answer box following each question, a yellow box with an arrow will appear on the right side of the box. Clicking this arrow button will open a pop-up window with with the MathQuill tools for inputting normal math symbols. 1 2 3 4 5 6 7 8 9 10 1 Fraction 6 Parentheses 2 Exponent 7 Pi 3 Square Root 8 Infinity 4 nth Root 9 Does Not Exist 5 Absolute Value 10 Touchscreen Tools Preview Button Next to the answer box, there will generally be a preview button. By clicking this button, the system will display exactly how it interprets the answer you keyed in. Remember, the preview button is not grading your answer, just indicating if the format is correct. Tips Tips will indicate the type of answer the system is expecting. Pay attention to these tips when entering fractions, mixed numbers, and ordered pairs. For additional help: www.xyzhomework.com/students ASCII Math (Calculator Style) Entry Some question types require a mathematical expression or equation in the answer box. Because XYZ Homework follows order of operations, the use of proper grouping symbols is necessary.
    [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]
  • Just-In-Time 22-25 Notes
    Just-In-Time 22-25 Notes Sections 22-25 1 JIT 22: Simplify Radical Expressions Definition of nth Roots p • If n is a natural number and yn = x, then n x = y. p • n x is read as \the nth root of x". p { 3 8 = 2 because 23 = 8. p { 4 81 = 3 because 34 = 81. p 2 • When n =p 2 we call it a \square root". However, instead of writing x, we drop the 2 and just write x. So, for example: p { 16 = 4 because 42 = 16. Domain of Radical Expressions • Howp do you find the even root of a negative number? For example, imagine the answer to 4 −16 is the number x. This means x4 = −16. The problem is, if we raise a number to the fourth power, we never get a negative answer: (2)4 = 2 · 2 · 2 · 2 = 16 (−2)4 = (−2)(−2)(−2)(−2) = 16 • Since there is no solution here for x, we say that the fourth root is undefined for negative numbers. • However, the same idea applies to any even root (square roots, fourth roots, sixth roots, etc) p If the expression contains n B where n is even, then B ≥ 0 . Properties of Roots/Radicals p • n xn = x if n is odd. p • n xn = jxj if n is even. p p p • n xy = n x n y (When n is even, x and y need to be nonnegative.) p q n x n x p • y = n y (When n is even, x and y need to be nonnegative.) p p m • n xm = ( n x) (When n is even, x needs to be nonnegative.) mp p p • n x = mn x 1 Examples p p 1.
    [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]
  • Study Guide Real Numbers
    REAL NUMBERS What Are Real Numbers? Real numbers are the set of rational and irrational numbers. The set of rational numbers includes integers, whole numbers, and natural numbers. • A rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. • An irrational number is a number that cannot be made into a fraction. Decimals that do not repeat or end are irrational numbers. • A square root of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. o A square root does not have to be a whole number or a rational number. The square root of 1.44 is 1.2 and the square root of 3 is 1.732050808…. o There are two solutions for a square root of a positive number, positive and negative. Square roots are used in mathematical operations.Sign The InequationsSign are solvedUp using inverse operations and when x² is found, the square root of a number is taken to find the answer. o The square of a number is a number multiplied by itself. Three squared, 3², is equal to 3 · 3, which is 9. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathWorksheets.com. How to use real numbers Any real number is either rational or irrational. Pi is an irrational number. Which of the following numbers are rational? Which are irrational? Ex.
    [Show full text]