2025 Year 10 MAM Headstart Booklet PDF
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This document is a headstart booklet for Year 10 Mathematical Methods, covering the topics indices, surds, and logarithms. It includes exercises and a timeline for the upcoming term. The booklet is intended to be used in 2025.
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YEAR 10 MATHEMATICAL METHODS HEADSTART 2025 This booklet contains the exercises that will be covered during Headstart. This work will be checked by your teacher in Week 1 of Term 1, 2025. Please refer to the course outline over the page for course sequence and...
YEAR 10 MATHEMATICAL METHODS HEADSTART 2025 This booklet contains the exercises that will be covered during Headstart. This work will be checked by your teacher in Week 1 of Term 1, 2025. Please refer to the course outline over the page for course sequence and assessment for Term 1. The full course timeline will be released at the start of 2025. EXERCISES/QUESTIONS TO COMPLETE DURING HEADSTART EXERCISE CHAPTER QUESTIONS Number classification 1.2 1-12, 14, 15, 17, 20, 23, 25 re 1.3 1-6, 9, 11, 13, 14, 16, 18 vie 1.4 1-20ac, 21-23a, 24-35ad, 36, 39 w 1.5 1-12ad, 15-17ac, 18-20a, 22, 25, Su 1.6 27 1-16ac, 18, 21, 25, 27, 31 rd 1.7 1-22ac, 25-27a, 29, 31, 33 s 1.8 1-15b, 17, 21-22ac, 25, 26, 28 Operation with Surds Complete exercises not finish during Headstart Review of index laws Negative indices Fractional Indices Combining index laws Holiday Homework: 2025 10MAM Timeline (Headstart and Term 1) Date Topic Chapter k Events Headstart Week 1 25-Nov Number classification 1.2 review 26 1.3 S v Operation with Surds 1.4 v Review of index laws 1.5 29-N ov Week 2 2-De Negative indices 1.6 c 3-De Fractional Indices 1.7 c 4-De Combining index laws 1.8 c 5-De c 6-De c Term 1 Week 1 27-Ja Australia Day n 28-Ja Logarithms 1.1 Staff only day n 29-Ja Logarithm Laws 1.11 Staggered start day n 30-Ja Solving equations 1.12 n 31-Ja n Week 2 3-Fe Substitution 2.2 b 4-Fe Adding & subtracting 2.3 Music Recruitment Night b algebraic fractions 5-Fe Multiplying & dividing 2.4 Zooper Dooper Fundraiser b algebraic fractions 6-Fe Solving simple equations 2.5 Student Photos b 7-Fe Year 7 and 8 Swimming b Week 3 10-Feb Solving multi-step 2.6 equations b Literal equations 2.7 b Review 2.8 13-Fe Evacuation Drill (Period 3) b 14-Fe Senior (Year 11 and 12) Round b Robin Sports Week 4 17-Fe CAT: Skills TF (Indices and Surds Ch. 1 & Year 12 Camp b & Algebra and Equations) 2 18-Fe Sketching linear graphs 3.2 Year 12 Camp & Year 7 Parent b Information Evening 19-Fe Determining linear equations 3.3 Year 12 Camp b 20-Fe Parallel & perpendicular lines 3.4 Year 12 Camp & Year 11 b Respect Connect Day 21-Fe Year 12 Camp b Week 5 24-Feb Distance between two 3.5 points b Midpoint of a line segment 3.6 Catch up photo day b Applications of Collinearity 3.7 GWSC Clubs Day 27-Fe Graphical solution 4.2 Intermediate (Year 9 and 10) b Round Robin Sport 28-Fe b Week 6 3-Ma Algebraic solution by substitution 4.3 Division Swimming r 4-Ma Algebraic solution by elimination 4.4 r 5-Ma Applications of 4.5 r simultaneous equations 6-Ma r 7-Ma r Week 7 10-M Labour Day ar 11-M Solving simultaneous linear & 4.6 World’s Greatest Shave ar non linear equations r Solving linear inequalities 4.7 NAPLAN Writing r Inequalities on the Cartesian 4.8 NAPLAN Reading 14-M NAPLAN Conventions of Languag ar Week 8 r Review 3.8 NAPLAN Numeracy r Review 4.1 NAPLAN Catch up r Review NAPLAN Catch up r Review NAPLAN catch up 21-M NAPLAN catch up ar Senior House Athletics Week 9 24-M CAT: Modelling TA (Linear Ch. 3 & ar Graphs and Simultaneous 4 - 4.5 Equations) 25-M Pythagoras’ Theorem 5.2 Year 7 Round Robin Sport ar 26-M Pythagoras’ Theorem in 3-D 5.3 ar r Trigonometric ratios 5.4 Year 9 OED Hike r Calculating side lengths 5.5 Regional Swimming & Year 9 OED Week 10 r Calculating angle size 5.6 Year 7 Camp 1-Ap Angles of elevation & depression 5.7 Year 7 Camp & Year 8 Round Rob r 2-Ap Bearings 5.8 Year 7 Camp r 3-Ap Year 7 Camp r 4-Ap Year 7 Camp r 1 Indices, surds and logarithms LEARNING SEQUENCE 1.1 Overview....................................................................................................................................................................2 1.2 Number classification review.............................................................................................................................4 1.3 Surds (10A)...............................................................................................................................................................9 1.4 Operations with surds (10A)............................................................................................................................14 1.5 Review of Index laws.........................................................................................................................................28 1.6 Negative indices...................................................................................................................................................35 1.7 Fractional indices (10A).....................................................................................................................................40 1.8 Combining index laws.......................................................................................................................................47 1.9 Application of indices: Compound interest...............................................................................................53 1.10 Logarithms (10A)..................................................................................................................................................60 1.11 Logarithm laws (10A)..........................................................................................................................................66 1.12 Solving equations (10A)....................................................................................................................................73 1.13 Review.....................................................................................................................................................................79 1.1 Overview Why learn this? We often take for granted the amount of time and effort that has gone into developing the number system we use on a daily basis. In ancient times, numbers were used for bartering and trading goods between people. Thus, numbers were always attached to an object; for example, 5 cows, 13 sheep or 20 gold coins. Consequently, it took a long time before more abstract concepts such as the number 0 were introduced and widely used. It took even longer for negative numbers or irrational numbers such as surds to be accepted as their own group of numbers. Historically, there has always been resistance to these changes and updates. In folk law, Hippasus — the man first credited with the discovery of irrational numbers — was drowned at sea for angering the gods with his discovery. A good example of how far we have come is to look at an ancient number system most people are familiar with: Roman numerals. Not only is there no symbol for 0 in Roman numerals, but they are extremely clumsy to use when adding or subtracting. Consider trying to add 54 (LIV) to 12 (XII). We know that to determine the answer we add the ones together and then the tens to get 66. Adding the Roman numeral is more complex; do we write LXVIII or LIVXII or LVXI or LXVI? Having a better understanding of our number system makes it easier to understand how to work with concepts such as surds, indices and logarithms. By building our understanding of these concepts, it is possible to more accurately model real-world scenarios and extend our understanding of number systems to more complex sets, such as complex numbers and quaternions. Where to get help Go to your learnON title at www.jacplus.com.au to access the following digital resources. The Online Resources Summary at the end of this topic provides a full list of what’s available to help you learn the concepts covered in this topic. Fully worked Interactivities solutions to every question Video Digital eWorkbook eLessons documents 2 Jacaranda Maths Quest 10 + 10A Exercise 1.1 Pre-test Complete this pre-test in your learnON title at www.jacplus.com.au and receive automatic marks, immediate corrective feedback and fully worked solutions. 1. Positive numbers are also known as natural numbers. Is this statement true or false? √ 2. State whether 36 is a rational or irrational number. 15 3. Simplify the following: 3n × 5n13. 5 10 15 4. Simplify the following: √ 32p q. −34 5. Determine the exact value of 81. MC √ 2 6. Select which of the numbers of the set { 0.25, , 0.261, −5, 3} are rational. {√ 2 A. 0.25, , 0.261} {0.261, −5, 3 B. } C. { , 0.261} D. } {√ √ 2 3 { 0.25, 0.261, −5, 3 9x10 × x simplifies to: MC 7. 12x8 × 3x7 E. 0.25, , 0.261, −5} 2 26 26 A. 4x B. 4x C.5x 5x2 2 D.x 4 E. 3 3 √ √ 8. Simplify the following expression: 3 2 × 10. √ √ √ 9. Simplify the following expression: 5 2 + 12 2 − 3 2. MC √ 3 √ √ 5 10. Choose the most simplified form of the following expression: 8a + 18a + a A. a √ 2 √ 4 2a 2a + 3 2a + a √ √ 5 2a + a√ B. a 2√ √ 2 C. a 2a 2a + 2 3a + a √ 2√ √ 4 1 y: 125= 5y+2. 2a 2a + 2 3a + a √ √ √ 2 D. a 2a 2a + 3 2a + a √ E. a 11. Solve the following equation for equation for x: x = log1 4 16. 12. Solve the following 13. Calculate the amount of interest earned on an investment of $3000 compounding annually at 3%p. a. for 3 years, correct to the nearest cent. log2 ) 14. Simplify the ( + log2(32) − log2(8). 14 following expression. MC 15. Choose the correct value for x in 3 + log23 = log2x. A. x = 0 B. x = 3 C. x = 9 D. x = 24 E. x = 27 TOPIC 1 Indices, surds and logarithms 3 At the end of this subtopic you should be able to: define the real, rational, irrational, integer and natural numbers determine whether a number is rational or irrational. 1.2.1 The real number system The number systems used today evolved from a basic and eles-4661 practical need of primitive people to count and measure 1.2 Number classification magnitudes and quantities such as livestock, people, review possessions, time and so on. As societies grew and architecture and engineering developed, number systems became more sophisticated. Number use developed from LEARNING INTENTION solely whole numbers to fractions, decimals and irrational numbers. The real number system contains the set of rational and irrational numbers. It is denoted by the symbol R. The set of real numbers contains a number of subsets which can be classified as shown in the chart below. Real numbers R Irrational numbers I (surds, non-terminating and non-recurring decimals, π, e) Rational numbers Q Non-integer rationals Integers Z Zero decimals) Positive Z+ (terminating and recurring – (Natural numbers N)Negative Z (neither positive nor negative) 4 Jacaranda Maths Quest 10 + 10A Integers (Z) The set of integers consists of whole positive and negative numbers and 0 (which is neither positive nor negative). The set of integers is denoted by the symbol Z and can be visualised as: Z = {… , −3, −2, −1, 0, 1, 2, 3, …} The set of positive integers are known as the natural numbers (or counting numbers) and is denoted Z+ or N. That is: Z+ = N = {1, 2, 3, 4, 5, 6, …} The set of negative integers is denoted Z−. Z− = {… − 6, −5, −4, −3, −2, −1} Integers may be represented on the number line as illustrated below. or natural numbers Z –6 ––5 –4 –3 –2 –1 The set of negative –3 –2 –1 0 1 2 3 Z The set of integers Rational numbers (Q) integers 1 2 3 4 5 6 N The set of positive integers a A rational number is a number that can be expressed as a ratio of two integers in the form b, where b ≠ 0. The set of rational numbers are denoted by the symbol Q. Rational numbers include all whole numbers, fractions and all terminating and recurring decimals. Terminating decimals are decimal numbers which terminate after a specific number of digits. Examples are: 5 9 4= 0.25, 8= 0.625, 5= 1.8. 1 Recurring decimals do not terminate but have a specific digit (or number of digits) repeated in a pattern. Examples are: 1 ̇ 3= 0.333 333 … = 0.3 or 0.3 133 ̇ ̇ 666= 0.199 699 699 6 … = 0.199 6 or 0.1996 Recurring decimals are represented by placing a dot or line above the repeating digit/s. Using set notations, we can represent the set of rational numbers as: a Q = { b∶ a, b ∈ Z, b ≠ 0} a This can be read as ‘Q is all numbers of the form bgiven a and b are integers and b is not equal to 0’. TOPIC 1 Indices, surds and logarithms 5 Irrational numbers (I) a An irrational number is a number that cannot be expressed as a ratio of two integers in the form b, where b ≠ 0. All irrational numbers have a decimal representation that is non-terminating and non-recurring. This means the decimals do not terminate and do not repeat in any particular pattern or order. For example:√ 5 = 2.236 067 997 5 … = 3.141 592 653 5 … e = 2.718 281 828 4 … The set of irrational numbers is denoted by the symbol I. Some common irrational numbers that you may √ √ be familiar with are 2, , e, 5. The symbol (pi) is used for a particular number that is the circumference of a circle whose diameter is 1 unit. In decimal form, has been calculated to more than 29 million decimal places with the aid of a computer. Rational or irrational Rational and irrational numbers combine to form the set of real numbers. We can find all of these number somewhere on the real number line as shown below. 2 –. –4 –3.236 –√3 2eπ –0.1 3 –5 –4 –3 –2 –1 0 1 2 3 4 5 R To classify a number as either rational or irrational: 1. Determine whether it can be expressed as a whole number, a fraction, or a terminating or recurring decimal. 2. If the answer is yes, the number is rational. If no, the number is irrational. WORKED EXAMPLE 1 Classifying numbers as rational or irrational Classify whether the following numbers are rational or irrational. √ a. e. 0.54 √3 b. 25 √ 1 f. 64 √3 √ 1 c. 13 d. 3 5 3 g. 32 h. 27 THINK WRITE 1 1 √ a. 5is already a rational number. a. 5is rational. b. 1. Evaluate 25. √ b. 25 = 5 √ √ √ 2. The answer is an integer, so classify 25. 25 is rational. c. 1. Evaluate 13. c. √ 13 = 3.605 551 275 46 … 2. The answer is a non-terminating 6 Jacaranda Maths Quest 10 + 10A and non-recurring decimal; classify √ 13 is irrational. √ 13. d. 1. Use your calculator to find the value of 3. d. 3 = 9.424 777 960 77 … 2. The answer is a non-terminating 3 is irrational. e. 0.54 is and non-recurring decimal; classify 3. e. 0.54 is a terminating rational. decimal; classify it accordingly. 3 3 f. 1. Evaluate √ 64. f. √ 64 = 4 3 2. The answer is a whole number, √ 64 is rational. 3 so classify √ 64. 3 3 g. 1. Evaluate √ 32. g. √ 32 = 3.17480210394 … 3 √ 2. The result is a non-terminating √ 32 is irrational. 1 and non-recurring decimal; classify 3 √ √ 32. 1 3 1 h. 1. Evaluate 3 27. h. 27= 3. √ 1 2. The result is a number in a Resourceseses rational form. 3 27is rational. Resources eWorkbook Topic 1 Workbook (worksheets, code puzzle and project) (ewbk-2027) Interactivities Individual pathway interactivity: Number classification review (int-8332) The number system (int-6027) Recurring decimals (int-6189) Exercise 1.2 Number classification review Individual pathways PRACTISE CONSOLIDATE MASTER 1, 4, 7, 10, 13, 14, 17, 20, 23 2, 5, 8, 11, 15, 18, 21, 24 3, 6, 9, 12, 16, 19, 22, 25 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency For questions 1 to 6, classify whether the following numbers are rational (Q) or irrational (I). 4 2. WE1 1. √ a. 4 5 a. 7√ 1 b. 0.04 2 2 c. c. 7 b. 9 d. 2 √ d. 5 √ √ a. b. 0.15 c. −2.4√ 9 3. 4 d. 100 TOPIC 1 Indices, surds and logarithms 7 √ 25 a. 14.4√ √ 9 4. b. 1.44 c. d. 5. a. 7.32 −√ b. 21 √ c. 1000 d. 7.216 349 157 … √ 1 a. 81 b. 3 √3 16 6. −√ c. 62 d. For questions 7 to 12, classify the following numbers as rational (Q), irrational (I) or neither. √ 1 0 7. 8 a. c. 8 d. 11 b. 625 4 √ 1.44 b. 81 −√ 1 8. −6 7 a. 9. √ d. c. 11 √3 √3 4 8 c. 21 7 a. 0 d. b. √ √ 1 3 64 a. − 11 c. 16 3 10. (−5)2 b. d. 100 a. √ √ 62 √3 1√ 2 25 b. c. 27 4 d. 11. b. −1.728 6√ 7 12.22 c. 4 4√ d. 6 √3 a. MC 13. Identify a rational number from the following. √ √ 4 9 D. 3√ A. 12 B. C. √3 E. 5 9 MC 14. Identify which of the following best represents an irrational number from the following numbers. C. 343 √ 6 −√ A. 81 5 B. D. 22 √ E. 144 √3 MC √ √ 15. Select which one of the following statements regarding the numbers −0.69, 7, 3, 49 is correct. A. 3is the only rational number. √ √ 7 and 49 are both irrational numbers. B. √ C. −0.69 and 49 are the only rational numbers. D. −0.69 is the only rational number. E. rational number. √ 7 is the only MC 1 11 √ 3 16. Select which one of the following statements regarding the numbers 2 2, − 3, 624,√ 99 is correct. 11 √ A. − 3and 624 are both irrational numbers. √ 3 624 is an irrational number and √ 99 is a rational number. B. √ 3 624 and √ 99 are both irrational numbers. C. 1 11 D. 2 2is a rational number and − 3is an irrational number. E. rational number. 3 √ 99 is the only 8 Jacaranda Maths Quest 10 + 10A Understanding √ 2 a 17. Simplify 2 b. MC √ 18. If p < 0, then p is: MC √ 2 A. positive B. negative C. rational D. irrational E. none of these 19. If p < 0, then p must be: A. positive B. negative C. rational D. irrational E. any of these Reasoning √ √ ) √ √ ) 20. Simplify ( p − q ×( p + q. Show full working. 2 2 2 21. Prove that if c = a + b , it does not follow that a = b + c. √ 2 22. Assuming that x is a rational number, for what values of k will the expression x + kx + 16 always be rational? Justify your response. Problem solving 36 23. Determine the value of m and n if 11is written as: a. 3 +1 b. 3 +1 c. 3 +1 d. 3 +1m n 3 +1mn 3 +1 1+mn 3 +mn 3 4 ,√ ,√. √ √ 3 4 Examples of surds include: 7, 5,√ 11,√ 15. √ √ 3 4 The numbers 9, 16,√ 125, and √ 81 are not surds as √ they can be simplified to rational numbers, that is: 9 = √ 3 4 3, 16 = 4,√ 125 = 5,√ 81 = 3. eles-4662 −1 1 −1 −1 24. If x means x, determine the value of 3 − 4 −1 3−1 + 4. −n 1 n −n −n 25. If x = x , evaluate 3 − 4 −n 3−n + 4 when n = 3. 1.3 Surds (10A) LEARNING INTENTION At the end of this subtopic you should be able to: determine whether a number under a root or radical sign is a surd prove that a surd is irrational by contradiction. 1.3.1 Identifying surds TOPIC 1 Indices, surds and logarithms 9 √A surd is an irrational number that is represented by a root sign or a radical sign, for example: WORKED EXAMPLE 2 Identifying surds Determine which of the following numbers are surds. √ √ 1 √3 e. 63 √3 a. 16 √ b. 13 c. d. 17 √4 f. 1728 16 √ √ THINK WRITE a. 1. Evaluate 16. a. 16 = 4 conclusion. 2. The answer is rational (since it is a √ whole number), so state your 16 is not a surd. √ √ b. 1. Evaluate 13. b. 13 = 3.605 551 275 46 … √ √ 2. The answer is irrational (since it is 13 is a surd. 1 a non-recurring and non-terminating decimal), so state your conclusion. √ 1 c. 1. Evaluate 1 16. c. 16= 4 2. The answer is rational (a fraction); 16is not a surd. state your conclusion. √ 1 3 3 d. 1. Evaluate √ 17. d. √ 17 = 2.571 281 590 66 … 3 2. The answer is irrational (a √ 17 is a surd. non-terminating and non-recurring decimal), so state your conclusion. 4 4 e. 1. Evaluate √ 63. e. √ 63 = 2.817 313 247 26 … 4 2. The answer is irrational, so √ 63 is a surd. 4 classify √ 63 accordingly. types of proofs. One such method is called proof by contradiction. This proof is so named because the logical argument of the proof is based on an assumption that leads to contradiction within the proof. Therefore the original assumption must be false. An irrational number is one that cannot be expressed in eles-4663 a 3 3 the form b(where a and b are integers). The next worked f. 1. Evaluate √ 1728. f. √ 1728 = 12 √ example sets out to prove that 2 is irrational. 3 2. The answer is rational; state your conclusion. √ 1728 is not a surd. So b, d and e are surds. 1.3.2 Proof that a number is irrational 10 Jacaranda Maths Quest 10 + 10A In Mathematics you are required to study a variety of √ WORKED EXAMPLE 3 Proving the irrationality of 2 √ Prove that 2 is irrational. THINK WRITE √ √ a 1. Assume that 2 is rational; that is, it can be Let 2 = b, where a and b are integers that have no a common factors and b ≠ 0. written as bin simplest form. We need to show that a and b have no common factors. 2 2 2. Square both sides of the equation. 2 =a b 3. Rearrange the equation to formula. 2 make a the subject of the a2 = 2b2 2 2 2 2 4. 2b is an even number and 2b = a. ∴ a is an even number and a must also be even; that is, a has a factor of 2. 5. Since a is even it can be written as a = 2r. ∴ a = 2r 2 2 6. Square both sides. a = 4r But a2 = 2b2from 2 2 7. Equate and. ∴ 2b = 4r b2 =4r2 2 = 2r2 ∴ b2is an even number and b must also be even; that is, b has a factor of 2. √ a where a and b have no common factors. 8. Use reasoning to deduce that 2 = bwhere a √ and b have no common factor. ∴ 2 is not rational. ∴ It must be irrational. Both a and b have a common factor of 2. This √ a contradicts the original assumption that 2 = b, Note: An irrational number written in surd form gives an exact value of the number; whereas the same number written in decimal form (for example, to 4 decimal places) gives an approximate value. DISCUSSION √ How can you be certain that root a is a surd? Resources Resourceseses eWorkbook Topic 1 Workbook (worksheets, code puzzle and project) (ewbk-2027) Digital document SkillSHEET Identifying surds (doc-5354) Interactivity Surds on the number line (int-6029) TOPIC 1 Indices, surds and logarithms 11 Exercise 1.3 Surds (10A) Individual pathways CONSOLIDATE 2, 5, 9, 12, 15, MASTER PRACTISE 18 3, 6, 10, 13, 16, 19 1, 4, 7, 8, 11, 14, 17 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency WE2 For questions 1 to 6, determine which of the following numbers are surds. b. 48 √ a. 81 √ c. 16 √ d. 1.6 √ 1. √ 3 3 c. a. 0.16 √ √ √ 3 b. 11 2. d. 4 27 a. 1000 √ √ d. 10 3. a. 32 √ b. 1.44 4√ c. 100 √3 c. 100 2 +√ 4. √3 √ √ d. 125 7 6 +√ b. 361 √3 a. 6 b. 2 √3 c. 169 5. d. 8 a. 16 b. (√ √3 c. 33 √ d. 0.0001 6. 7)2 √4 e. 32 √ √5 f. 80 √ 6 MC 7. The correct statement regarding the set √ √ 3 √ of numbers { 9, 20, 54 ,√ 27, 9} is: 3 √ √ 27 and 9 are the only rational numbers of the set. A. √ 6 9is the only surd of the set. B. √ 6 √ 9and 20 are the only surds of the set. C. √ √ 20 and 54 are the only surds of the set. D. √ √ 9 and 20 are the only surds of the set. E. √ √ √ 1 1 1 MC √ 8. Identify the √ numbers from the set 1 27, 1 { √ 3 4and √ 8, 21,√ 8} that are √ E. 21 only √ surds. 1 A. 21 only B. only √ 1 8and √ 8and √3 12 Jacaranda Maths D. 21 only Quest 10 + 10A C. 8 8 3 4, √ MC 1 9. Select a statement regarding the set of √ √ √ numbers { , 49, 12, 16, 3, +1} that is not true. √ A. 12 is a surd. 12 and √ √ B. 16 are surds. √ C. is irrational but not a surd. 12 and √ D. 3 + 1 are not rational. E. is not a surd. MC √ √ 10. Select a statement regarding the set of numbers {6 7, 144 √ √ √ √ √ 144 16, 7 6, 9 2, 18, 25} that is not true. √ 144 B. 25 are not surds. A. when simplified is an integer. 16 16and √ √ C. 2. 9 2 is smaller than 6√ √ 7 6 is smaller than 9√ D. 7. √ E. 18 is a surd. Understanding √ 11. Complete the following statement by selecting appropriate words, suggested in brackets: a is definitely not a surd, if a is… (any multiple of 4; a perfect square; cube). 12. Determine the 3 smallest value of m, where m is a positive integer, so that √ 16m is not a surd. 13. a. Determine any combination of m and n, where m and n are positive integers with m < n, so that 4 √ (m + 4) (16 − n) is not a surd. b. If the condition that m < n is removed, how many possible combinations are there? Reasoning 14. Determine whether the following are rational or √ (√ √ ) (√ √ irrational. a. 2 5 −√ 5 + 2 5 − 2) √ 5 +√ b. 2 c. WE3 15. Prove that the following numbers are irrational, using a proof by contradiction: a. 3√ √ b. 5√ c. 7. √ ( √ )( 16. is an irrational number and so is 3. Therefore, determine whether − 3 + √ ) 3 is an irrational number. Problem solving 17. Many composite numbers have a variety of factor pairs. For example, factor pairs of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6. a. Use each pair of possible factors to simplify the following surds. i. 48 √ √ ii. 72 b. Explain if the factor pair chosen when simplifying a surd affect the way the surd is written in simplified form. c. Explain if the factor pair chosen when simplifying a surd affect the value of the surd when it is written in simplified form. √ √ √ √ 18. Consider the expression ( p + q)( m − n). Determine under what conditions will the expression produce a rational number. √ √ √ 19. Solve 3x − 12 = 3 and indicate whether the result is rational or irrational. TOPIC 1 Indices, surds and logarithms 13 Simplification of a surd uses the method of multiplying surds in reverse. The process is summarised in the following steps: 1. Split the number under the radical into the product of two factors, one of which is a perfect square. 2. Write the surd as the product of two surds multiplied together. The two surds must correspond to the factors identified in step 1. 3. Simplify the surd of the perfect square and write the surd √ in the form a b. eles-4664 √ The example below shows the how the surd 45 can be 1.4 Operations with surds √ √ simplified by following the steps 1 to 3. 45 = 9 × 5 (Step (10A) 1) LEARNING INTENTION At the end of this subtopic you should be able to: multiply and simplify surds add and subtract like surds divide surds rationalise the denominator of a fraction. 1.4.1 Multiplying and simplifying surds Multiplication of surds To multiply surds, multiply the expressions under the radical sign. √ √ √ √ For example: 8 × 3 = 8 × 3 = 24 If there are coefficients in front of the surds that are being multiplied, multiply the coefficients and then multiply the expressions under the radical signs. √ √ √ √ For example: 2 3 × 5 7 = (2 × 5) 3 × 7 = 10 21 Multiplication of surds In order to multiply two or more surds, use the following: √ √ √ a× b= a×b √ √ √ m a × n b = mn a × b where a and b are positive real numbers. Simplification of surds To simplify a surd means to make the number under the radical sign as small as possible. Surds can only be simplified if the number under the radical sign has a factor which is a perfect square (4, 9, 16, 25, 36, …). √ √ = 3 × 5 = 3 5 (Step 3) If possible, try to factorise the number under the radical sign so that the largest possible perfect square is used. This will ensure the surd is simplified in 1 step. 14 Jacaranda Maths Quest 10 + 10A √ √ = 9 × 5 (Step 2) Simplification of surds √ √ 2 n= a ×b √ 2 √ = a × b √ =a× b √ =a b WORKED EXAMPLE 4 Simplifying surds Simplify the following surds. Assume that x and y are positive real 1 numbers. b. 405 − 8√ a. 384 3√ √ √ 3 5 c. 175 5 180x y d. THINK WRITE factor is the largest possible perfect a. 1. Express 384 as a product of two square. factors where one √ √ 384 = 64 × 6 a. √ √ √ √ 2. Express 64 × 6 as the product of two surds. = 64 × 6 square (that is, 64 √ = 8).= 8 6 3. Simplify the square root from the perfect possible perfect square. √ √ b. 1. Express 405 as a product of two b. 3 405 = 3 81 × 5 factors, one of which is the largest √ √ √ √ 2. Express 81 × 5 as a product of two surds. = 3 81 × 5 3. Simplify 81. = 3 × √ 9 5 possible perfect square. 4. Multiply together the whole numbers √ outside the square root sign (3 and 9). = 27 5 c. 1. Express 175 as a product of two 1 √ 1 √ c. − 8 175 = − 8 25 × 7 factors in which one factor is the largest √ 1 √ √ √ 1 √ 2. Express 25 × 7 as a product of 2 surds. = − 8× 25 × 7 3. Simplify 25. = − 8× 5 7 √ 3 5 √ 2 4 4. Multiply together the numbers outside the d. 5 180x y = 5 36 × 5 × x × x × y × y = square root sign. 3 5 d. 1. Express each of 180, x and y as a product of √ 2 4 √ 5 × 36x y × 5xy two factors where one factor is the largest possible perfect square. 2. Separate all perfect squares into one surd and all other factors into the other surd. TOPIC 1 Indices, surds and logarithms 15 5 √ =− 8 7 √ 2 4 2 √ 3. Simplify 36x y. = 5 × 6 × x × y × 5xy surds 4. Multiply together the numbers and the 2√ pronumerals outside the square root sign. = 30xy 5xy WORKED EXAMPLE 5 Multiplying Multiply the following surds, expressing answers in the simplest form. Assume that x and y are positive real numbers. √ √ 11 ×√ b. 5 6 12 × 2√ √ a. 7 5 3 × 8√ √ 5 2 √ 2 c. 6 15x y × 12x d. y THINK WRITE a. Multiply the surds together, using b. Multiply the coefficients together and then multiply the surds together. a. √ √ √ √ √ √ √ a × b = ab (that is, multiply b. 5 3 × 8 5 = 5 × 8 × 3 × 5 = 40 expressions under the square root √ sign). × 3×5 Note: This expression cannot be √ = 40 15 simplified √ √ √ √ 11 × 7 = 11 × 7 = 77 any further. √ √ √ √ √ √ √ c. 1. Simplify 12. c. 6 12 × 2 6 = 6 4 × 3 × 2 6 = 6 × 2 3 × 2 6 √ √ = 12 3 × 2 6 √ = 24 18 2. Multiply the coefficients together and multiply the surds together. √ √ 3. Simplify the surd. = 24 9 × 2 = 24 × 3 2 √ = 72 2 √ 5 2 √ 2 √ 4 2 √ 2 d. 1. Simplify each of the surds. d. 15x y × 12x y = 15 × x × x × y × 4 × 3 × x × y √ √ = x2 × y × 15 × x × 2 × x × 3 × y √ √ = x2y 15x × 2x 3y √ = 2x3y 9 × 5xy 2. Multiply the coefficients together and the surds together. √ √ = x2y × 2x 15x × 3y = 2x3y 45xy 3 √ 3 √ 3. Simplify the surd. = 2x y × 3 5xy = 6x y 5xy 16 Jacaranda Maths Quest 10 + 10A When working with surds, it is sometimes necessary to multiply surds by themselves; that is, square them. Consider the following examples: (√ 2 √ √ √ 2) = 2 × 2 = 4 = 2 (√ 2 √ √ √ 5) = 5 × 5 = 25 = 5 Observe that squaring a surd produces the number under the radical sign. This is not surprising, because squaring and taking the square root are inverse operations and, when applied together, leave the original unchanged. Squaring surds When a surd is squared, the result is the expression under the radical sign; that is: (√ 2 a) = a where a is a positive real number. WORKED EXAMPLE 6 Squaring surds Simplify each of the following. (√ √ ( 6)2 3 5)2 a. b. THINK WRITE (√ 2 √ a. Use a) = a to square 5. b. (√ 2 (√ 2 ( √ 2 2 (√ a) = a, where a = 6. a. 6) = 3 5) = 3 × 5)2 = 9 × 5 6 b. 1. Square 3 and apply subtracting surds is similar to adding and subtracting like terms in algebra. eles-4665 2. Simplify. = 45 1.4.2 Addition and subtraction of surds Surds may be added or subtracted only if they are alike. √ √ √ Examples of like surds include 7, 3 7 and − 5 7. √ √ √ √ Examples of unlike surds include 11, 5, 2 13 and − 2 3. In some cases surds will need to be simplified before you decide whether they are like or unlike, and then addition and TOPIC 1 Indices, surds and logarithms 17 subtraction can take place. The concept of adding and WORKED EXAMPLE 7 Adding and subtracting surds Simplify each of the following expressions containing surds. Assume that a and b are positive real numbers. √ √ √ a. 6 5 3 + 2 12 − 5 2 + 3√ √ √ 2 3 6 + 17 6 − 2√ √ √ √ 100a3b2 + ab 36a − 5 4a2 c. b b. 8 1 THINK WRITE √ a. 3√ √ √ √ the same surd ( 6). Simplify. 6 + 17 6 − 2 6 = (3 + 17 − 2) 6 a. All 3 terms are alike because they contain √ = 18 6 √ √ √ √ √ √ √ b. 1. Simplify surds where possible. b. 5 3 + 2 12 − 5 2 + 3 8 = 5 3 + 2 4 × 3 − 5 2 + √ 3 4×2 √ √ √ √ =5 3+2×2 3−5 2+3×2 2 √ √ √ √ √ √ 2=9 3+ 2 2. Add like terms to obtain the simplified = 5 3 + 4 3 − 5 2 + 6 answer. 1 √ 3 2 √ √ 2 1 √ 2 2 √ c. 1. Simplify surds where possible. c. 2 100a b + ab 36a − 5 4a b = 2× 10 a × a × b + ab × 6 a − √ 5×2×a b 1 √ √ √ = 2× 10 × a × b a + ab × 6 a − 5 × 2 × a b √ 2. Add like terms to obtain the simplified 10a b answer. √ √ √ √ = 5ab a + 6ab a − 10a b = 11ab a − TI | THINK DISPLAY/WRITE CASIO | THINK DISPLAY/WRITE a–c. as: a–c. √ √ √ √ In a new document, on a √ √ √ √ a–c. 3 6 + 17 6 − 2 6 5 3 + Calculator page, 3 6 + 17 6 − 2 6 5 3 + On the Main screen, √ √ 2 12 − 5 2 complete the entry lines √ √ complete the entry lines a–c. 2 12 − 5 2 as: √ 3 2 √ √ 2 100a b + ab 36a− 5 4a b|a > 0 and b > 0 1 Press ENTER after each 2 √ √ √ √ simplify 5 3 + 2 12 − 5 2 + 3 8 = √ 2 √ √ +3 8 √ √ √ √ 9 3+ 2 √ 3 6 + 17 6 − 2 6 = 18 6 +3 8 √ √ 2 √ a × b × 36a− 5 4a b|a > ( 100a3b2+ 1 0|b > 0) entry. 1 Press EXE after each entry. √ √ √ √ √ + 2 12 − 5 2 + 3 8 2 √ √ √ √ √ 100a3b2 + ab 36a− 3 6 + 17 6 − 2 6 = 18 6 5 3 √ 2 √ 5 4a b = 11a b − 10a b 23 √ √ =9 3+ 2 √ √ 100a3b2 + ab 36a− 1 2 √ 2 √ 5 4a b = 11a b − 10a b 23 18 Jacaranda Maths Quest 10 + 10A expressions under the radical signs. eles-4666 1.4.3 Dividing surds Dividing surds To divide surds, divide the √ √ a ab √ b= where a and b are positive real numbers. When dividing surds it is best to simplify them (if possible) first. Once this has been done, the coefficients are divided next and then the surds are divided. √ m a √ m √a n b= n b WORKED EXAMPLE 8 Dividing surds Divide the following surds, expressing answers in the simplest form. Assume that x and y are positive real numbers. √ √ √ √ 55 b.9 88 48 36xy a. √ √ c. √ d. √ 9 11 5 3 6 99 25x y THINK WRITE √ √ √ √ a a 55 55 5 a. 1. Rewrite √ √ the fraction, b= 5= using b. a. simplified any further. 2. Divide the numerator by the √ denominator (that is, 55 by 5). = 11 Check if the surd can be the fraction, √ √ using b= 3= √ √ √ a a b. b. 48 3 b. 1. Rewrite √ 48 √ √ 2. Divide 48 by 3. = 16 3. Evaluate 16. = 4 √ √ a b. c.9 88 c. 1. Rewrite √ √ 9 √88 a 6 99= 6 99 surds, using √ b= 2. Simplify the fraction under the radical by dividing both numerator and denominator by 11. 9 √8 = 6 9 TOPIC 1 Indices, surds and logarithms 19 √ 3. Simplify surds. =9 × 2 2 6 × 3 in the denominator together. √ 4. Multiply the whole numbers in =18 2 18 the numerator together and those √ √ 9 11 √ 5. Cancel the common factor of 18. = 2 25x y =6 xy d. 1. Simplify each surd. d. √ 8 10 √ 5√ √ 5 x × x × y × y =6 xy 5x4y xy 36xy √ case xy.=6 5x4y5 denominators eles-4667 2. Cancel any common factors — in this 1.4.4 Rationalising If the denominator of a fraction is a surd, it can be changed into a rational number through multiplication. In other words, it can be rationalised. As discussed earlier in this chapter, squaring a simple surd (that is, multiplying it by itself) results in a rational number. This fact can be used to rationalise denominators as follows. Rationalising the denominator √ √ √ √ a a b ab √ √ √ b b= b× b= If both numerator and denominator of a fraction are multiplied by the surd contained in the denominator, the denominator becomes a rational number. The fraction takes on a different appearance, but its numerical value is unchanged, because multiplying the numerator and denominator by the same number is equivalent to multiplying by 1. WORKED EXAMPLE 9 Rationalising the denominator Express the following in their simplest form with a rational denominator. √ a.2 12 √ √ √ √ √ 6 17 − 3 14 7 3 54 √ c. 13 b. 20 Jacaranda Maths Quest 10 + 10A THINK WRITE √ √ 13 fraction 13 a. 1. Write the fraction. a. 2. √ 6 √ 13 Multiply both the numerator = = √ √ 13× 78 surd contained in the and denominator by the 13 denominator (in this case √ 13 √ 13). This has the same √ 6 effect as multiplying the because √ by 1, 13= 1. √ b. 1. Write the fraction. b.2 12 √ 3 54 √ 2 12 2. Simplify the surds. (This avoids dealing √ with large numbers.) 3×3 6 √ √ √ 3 54=2 4 × 3 =4 3 √ √ 3 9×6 9 6 √ =2 × 2 3 effect as multiplying the √ 6 fraction by 1, √ √ √ 3. Multiply both the =4 3 9 6× 6= 1. √ numerator and denominator √ √ 6 6 =4 18 9 × √ 6 by 6. This has the same because Note: We need to multiply only by the surd part of the √ √ denominator (that is, by 6 rather than by 9 6.) √ √ 4. Simplify 18. =4 9 × 2 9 × 6 √ =4 × 3 2 54 √ =12 2 54 √ 9 and denominator by 7. Use 5. Divide both the numerator √ √ grouping symbols (brackets) to 17 − 3 14 and denominator by 6 (cancel down). √ make it clear that the whole 7 √ √ √ c. 1. Write the fraction. c. numerator must be multiplied =( 17 − 3 14) 7× √ √ √ by 7. 7 7 √ 2. Multiply both the numerator =2 2 TOPIC 1 Indices, surds and logarithms 21 √ √ √ √ √ √ 3. Apply the Distributive Law in the numerator. 17 × 7 − 3 14 × 7 7 × 7 √ √ 119 − 3 98 a (b + c) = ab + ac = = 7 √ √ √ 4. Simplify 98. = 119 − 3 49 × 2 7 √ == 119 − √ √ 21 2 7 119 − 3 √ ×7 27 √ √ √ √ √ √ √ 6 − 11, a + b and a − b, 2 5 − 7 and 2 5 + 7. This fact is used to rationalise denominators containing a sum or a difference of surds. eles-4668 1.4.5 Rationalising denominators Using conjugates to rationalise the denominator using conjugate surds The product of pairs To rationalise the denominator that contains a sum or a of conjugate surds results in a rational number. difference of surds, multiply both numerator and √ denominator by the conjugate of the denominator. Examples of pairs of conjugate surds include 6 + 11 and Two examples are given below: 2. To rationalise the denominator of the 1. To rationalise the denominator of the fraction 1 fraction 1 √ √ a + b, multiply it by √ √ √ √ √ √ √ √ a − b a − b. a + b a + b. √ √ a − b, multiply it by A quick way to simplify the denominator is to use the difference of two squares identity: (√ √ ) (√ √ ) (√ 2 (√ a − b a + b = a) − b)2 =a−b WORKED EXAMPLE 10 Using conjugates to rationalise the denominator Rationalise the denominator and simplify the following. a.1 √ 4− 3 √ 3 23 b. √ √ + 3 6+ 22 Jacaranda Maths Quest 10 + 10A THINK WRITE a. 1. Write the fraction. a.1 √ 4− 3 √ √ (4 − 3)×(4 + 3) 2. Multiply the √ numerator and =1 (4 + 3) denominator by the conjugate of the denominator. √ (Note that(4 + 3) √ √ =4 + 3 (4 + 3)= 1). in the numerator and the difference of two squares identity in the denominator. √ (4)2 − ( 3)2 3. Apply the Distributive Law √ 4. Simplify. =4 + 3 16 − 3 √ =4 + 3 13 denominator. √ b. 1. Write the fraction. b. (Note that(3 − 3) √ (3 − 3)= 1.) √ √ 2. Multiply the numerator and 6+3 2 denominator by the conjugate of the √ √ √ √ 3+ 3 (3 + 3)×(3 − 3) (3 − 3) √ √ =( 6 + 3 2) i o n s 3. i M n u g l r t o i u p p l i y n t g h = e symbols in the numerator, and apply e the difference of two squares identity in x the denominator. p √ √ √ √ √ √ 2 6 × 3 + 6 × (− 3) + 3 2 × 3 + 3 2 × −( 3) (3) r √ 2 e − ( 3) s s √ √ √ √ 4. Simplify. =3 6 − 18 + 9 2 − 3 6 9 − 3 √ √ =− 18 + 9 2 6 √ √ =− 9 × 2 + 9 2 6 √ √ =−3 2 + 9 2 6 √ =6 2 6 √ = 2 TOPIC 1 Indices, surds and logarithms 23 TI | THINK DISPLAY/WRITE CASIO | THINK DISPLAY/WRITE a-b. √ On a Calculator page, 4− 3 as: ) complete the entry lines simplify a-b. 1 √ √ √ a-b. ( 6+3 23+ 3 On the Main screen, 4 −√ ) simplify complete the entry lines ( √3 as: √ 1 a-b. 6+3 2 √ Press EXE after each entry. 3+ 3 √ √ Press ENTER after each entry. 4 − 3=4 + 3 1 13 √ √ 6+3 2 √ √ 3 + 3= 2 √ √ 4 − 3=4 + 3 1 √ √ 13 6+3 2 √ √ 3 + 3= 2 Resources Resourceseses eWorkbook Topic 1 Workbook (worksheets, code puzzle and project) (ewbk-2027) Digital documents SkillSHEET Simplifying surds (doc-5355) SkillSHEET Adding and subtracting surds (doc-5356) SkillSHEET Multiplying and dividing surds (doc-5357) SkillSHEET Rationalising denominators (doc-5360) SkillSHEET Conjugate pairs (doc-5361) SkillSHEET Applying the difference of two squares rule to surds (doc-5362) Video eLessons Surds (eles-1906) Rationalisation of surds (eles-1948) Interactivities Addition and subtraction of surds (int-6190) Multiplying surds (int-6191) Dividing surds (int-6192) Simplifying surds (int-6028) Conjugate surds (int-6193) Exercise 1.4 Operations with surds (10A) Individual pathways CONSOLIDATE MASTER PRACTISE 2, 5, 8, 11, 13, 16, 19, 22, 25, 3, 6, 9, 14, 17, 20, 23, 26, 29, 1, 4, 7, 10, 12, 15, 18, 21, 24, 28, 31, 34, 37, 40 32, 35, 38, 41 27, 30, 33, 36, 39 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency WE4a For questions 1 to 3, simplify the following surds. √ √ a. 12 √ 2. c. 27 √ b. 112 √ b. 24 √ 1. c. 68 √ d. 180 a. 54 √ d. 125 24 Jacaranda Maths Quest 10 + 10A b. 162 √ a. 88 √ √ 3. c. 245 √ d. 448 WE4b,c For questions 4 to 6, simplify the following surds. 1 4. 2√ c. 48 7√ d. 392 c. 80 7√ 1 b. 90 9√ b. 192 9√ a. 8 8√ 5. −6√ 3 c. 135 10√ a. 75 −7√ 1 a. 162 4√ b. 80 16√ d. 175 d. 54 1 6. 9√ WE4d For questions 7 to 9, simplify the following surds. Assume that a, b, c, d, e, f, x and y are positive real numbers. √ 2 √ 2√ √ 16a a. 72a b. 90a2 c. b 338a d. 4 7. √ √ 5√ √ 338a3b3 a. 68a3b b. 125x6y4 c. 5 80x3y d. 2 8. √ 7 5 √ 7 91 9. 6 162c d a. 2 405c d b. 2√ 1 √ 11 11 c. 88ef 2 392e f d. WE5a 10. Simplify the following expressions containing surds. Assume that x and y are positive real numbers. √ 3 5 + 4√ √ √ a. 5 2 3 + 5 3 +√ b. 3 √ c. 3 6 11 − 2√ √ √ √ 8 5 + 3 3 + 7 5 + 2√ d. 11 11. Simplify the following expressions containing surds. Assume that x and y are positive real numbers. √ √ 7 2 + 9 2 − 3√ √ √ √ a. 2 9 6 + 12 6 − 17 6 − 7√ √ √ √ b. 6 12 3 − 8 7 + 5 3 − 10√ √ √ √ c. 7 2 x + 5 y + 6 x − 2√ d. y WE5b For questions 12 to 14, simplify the following expressions containing surds. Assume that a and b are positive real numbers. √ √ a. 300 125 − 150 +√ 12. √ b. 600 200 −√ √ √ c. 75 2 20 − 3 5 +√ √ √ 27 − 3 +√ d. 45 √ √ √ 13. 6 12 + 3 27 − 7 3 +√ √ √ √ a. 18 150 + 24 − 96 +√ b. 108 √ √ √ c. 100 5 11 + 7 44 − 9 99 + 2√ √ √ √ 3 90 − 5 60 + 3 40 +√ 14. d. 121 √ √ √ 2 30 + 5 120 + 60 − 6√ √ √ √ a. 135 6 ab − 12ab + 2 9ab + 3√ √ 1 √ 1 b. 27ab 98 + 3 48 + 3√ 1 √ 7 √ c. 12 8 32 − 6 18 + 3√ 1 d. 72 2 WE5c For questions 15 to 17, simplify the following expressions containing surds. Assume that a and b are positive real numbers. √ √ √ a. 32a 10 a − 15 27a + 8 12a + 14√ √ √ √ 15. 7 a − 8a + 8 9a −√ b. 9a √ 2 √ √ 2 c. 54ab 16 4a − 24a + 4 8a +√ √ √ 150ab + 96ab −√ d. 96a √ √ √ 1 √ 1 √ 1 8a3 + 72a3 − 98a3 a. 2 36a + 4 128a − 6√ b. 144a 16. √ √ √ √ √ 9a3 + 3a5 c. 6 a5b + a3b − 5 a5 d. b TOPIC 1 Indices, surds and logarithms 25 √ √ 2 √ 3 3 ?
