2025 Year 10 Mathematical Methods Headstart Booklet PDF

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 review 1.2 1-12, 14, 15, 17, 20, 23, 25 Surds 1.3 1-6, 9, 11, 13, 14, 16, 18 Operation with Surds 1.4 1-20ac, 21-23a, 24-35ad, 36, 39 Review of index laws 1.5 1-12ad, 15-17ac, 18-20a, 22, 25, 27 Negative indices 1.6 1-16ac, 18, 21, 25, 27, 31 Fractional Indices 1.7 1-22ac, 25-27a, 29, 31, 33 Combining index laws 1.8 1-15b, 17, 21-22ac, 25, 26, 28 Holiday Homework: Complete exercises not finish during Headstart  2025 10MAM Timeline (Headstart and Term 1) Date Topic Chapter Task Events Headstart 25-Nov Number classification review 1.2 26-Nov Surds 1.3 Week 1 27-Nov Operation with Surds 1.4 28-Nov Review of index laws 1.5 29-Nov 2-Dec Negative indices 1.6 3-Dec Fractional Indices 1.7 Week 2 4-Dec Combining index laws 1.8 5-Dec 6-Dec Term 1 27-Jan Australia Day 28-Jan Logarithms 1.1 Staff only day Week 1 29-Jan Logarithm Laws 1.11 Staggered start day 30-Jan Solving equations 1.12 31-Jan 3-Feb Substitution 2.2 Adding & subtracting algebraic 4-Feb fractions 2.3 Music Recruitment Night Week 2 Multiplying & dividing algebraic 5-Feb fractions 2.4 Zooper Dooper Fundraiser 6-Feb Solving simple equations 2.5 Student Photos 7-Feb Year 7 and 8 Swimming 10-Feb Solving multi-step equations 2.6 11-Feb Literal equations 2.7 12-Feb Review 2.8 Week 3 13-Feb Evacuation Drill (Period 3) Senior (Year 11 and 12) Round Robin 14-Feb Sports CAT: Skills TF (Indices and Surds & 17-Feb Algebra and Equations) Ch. 1 & 2 Year 12 Camp Year 12 Camp & Year 7 Parent 18-Feb Sketching linear graphs 3.2 Information Evening Week 4 19-Feb Determining linear equations 3.3 Year 12 Camp Year 12 Camp & Year 11 Respect 20-Feb Parallel & perpendicular lines 3.4 Connect Day 21-Feb Year 12 Camp 24-Feb Distance between two points 3.5 25-Feb Midpoint of a line segment 3.6 Catch up photo day 26-Feb Applications of Collinearity 3.7 GWSC Clubs Day Week 5 Intermediate (Year 9 and 10) Round 27-Feb Graphical solution 4.2 Robin Sport 28-Feb  3-Mar Algebraic solution by substitution 4.3 Division Swimming 4-Mar Algebraic solution by elimination 4.4 Applications of simultaneous Week 6 5-Mar equations 4.5 6-Mar 7-Mar 10-Mar Labour Day Solving simultaneous linear & non- 11-Mar linear equations 4.6 World’s Greatest Shave Week 7 12-Mar Solving linear inequalities 4.7 NAPLAN Writing 13-Mar Inequalities on the Cartesian plane 4.8 NAPLAN Reading 14-Mar NAPLAN Conventions of Language 17-Mar Review 3.8 NAPLAN Numeracy 18-Mar Review 4.1 NAPLAN Catch up 19-Mar Review NAPLAN Catch up Week 8 20-Mar Review NAPLAN catch up NAPLAN catch up 21-Mar Senior House Athletics CAT: Modelling TA (Linear Graphs Ch. 3 & 4 - 24-Mar and Simultaneous Equations) 4.5 25-Mar Pythagoras’ Theorem 5.2 Year 7 Round Robin Sport Week 9 26-Mar Pythagoras’ Theorem in 3-D 5.3 27-Mar Trigonometric ratios 5.4 Year 9 OED Hike 28-Mar Calculating side lengths 5.5 Regional Swimming & Year 9 OED Hike 31-Mar Calculating angle size 5.6 Year 7 Camp 1-Apr Angles of elevation & depression 5.7 Year 7 Camp & Year 8 Round Robin Week 10 2-Apr Bearings 5.8 Year 7 Camp 3-Apr Year 7 Camp 4-Apr Year 7 Camp  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 1 — #1 Indices, surds and 1 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  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 2 — #2 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 Video Interactivities solutions eLessons to every question Digital eWorkbook documents 2 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 3 — #3 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. 1 1 3. Simplify the following: 3n 5 × 5n 3. √ 5 4. Simplify the following: 32p10 q15. −3 5. Determine the exact value of 81 4. √ 2 6. MC Select which of the numbers of the set { 0.25, 𝜋, 0.261, −5, } are rational. 3 √ 2 A. { 0.25, 𝜋, 0.261} B. {0.261, −5, } C. {𝜋, 0.261} √ √ 3 2 D. { 0.25, 0.261, −5, } E. { 0.25, 𝜋, 0.261, −5} 3 12x8 × 3x7 7. MC simplifies to: 9x10 × x3 5x2 5x26 x2 A. B. 4x2 C. 4x26 D. E. 3 3 4 √ √ 8. Simplify the following expression: 3 2× 10. √ √ √ 9. Simplify the following expression: 5 2 + 12 2 − 3 2. √ √ √ 10. MC Choose the most simplified form of the following expression: 8a3 + 18a + a5 √ √ √ √ √ A. 5 2a + a a B. 2a 2a2 + 3 2a + a4 a √ √ √ √ √ √ C. 2a2 2a + 2 3a + a4 a D. 2a2 2a + 2 3a + a2 a √ √ √ E. 2a 2a + 3 2a + a2 a 1 11. Solve the following equation for y: = 5y+2. 125 12. Solve the following equation for x: x = log 1 16. 4 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. ( ) 1 14. Simplify the following expression. log2 + log2 (32) − log2 (8). 4 15. MC Choose the correct value for x in 3 + log2 3 = log2 x. A. x = 0 B. x = 3 C. x = 9 D. x = 24 E. x = 27 TOPIC 1 Indices, surds and logarithms 3  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 4 — #4 1.2 Number classification review LEARNING INTENTION 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 eles-4661 The number systems used today evolved from a basic and practical need of primitive people to count and measure magnitudes and quantities such as livestock, people, possessions, time and so on. As societies grew and architecture and engineering developed, number systems became more sophisticated. Number use developed from 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 Rational numbers Q non-recurring decimals, π, e) Non-integer rationals Integers Z (terminating and recurring decimals) Positive Z+ Zero (neither positive nor negative) Negative Z– (Natural numbers N) 4 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 5 — #5 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. –3 –2 –1 0 1 2 3 Z 1 2 3 4 5 6 N Z – –6 –5 –4 –3 –2 –1 The set of integers The set of positive integers The set of negative integers or natural numbers Rational numbers (Q) a A rational number is a number that can be expressed as a ratio of two integers in the form , where b ≠ 0. b 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: 1 5 9 = 0.25, = 0.625, = 1.8. 4 8 5 Recurring decimals do not terminate but have a specific digit (or number of digits) repeated in a pattern. Examples are: 1 = 0.333 333 … = 0.3̇ or 0.3 3 133 ̇ 6̇ or 0.1996 = 0.199 699 699 6 … = 0.199 666 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 = { ∶ a, b ∈ Z, b ≠ 0} b a This can be read as ‘Q is all numbers of the form given a and b are integers and b is not equal to 0’. b TOPIC 1 Indices, surds and logarithms 5  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/10/9 — 6:09 — page 6 — #6 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 … 𝜋 = 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, 𝜋, e, 5. The symbol 𝜋 (pi) is used for a particular number that is the circumference of a circle whose diameter is 1 unit. In decimal form, 𝜋 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 –0.1 3 2 e π –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. 1 √ √ a. b. 25 c. 13 d. 3𝜋 5 √ √ 3 √3 1 3 e. 0.54 f. 64 g. 32 h. 27 THINK WRITE 1 1 a. is already a rational number. a. is rational. 5 5 √ √ 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 and 13 is irrational. √ non-recurring decimal; classify 13. 6 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 7 — #7 d. 1. Use your calculator to find the value of 3𝜋. d. 3𝜋 = 9.424 777 960 77 … 2. The answer is a non-terminating and 3𝜋 is irrational. non-recurring decimal; classify 3𝜋. e. 0.54 is a terminating decimal; classify it e. 0.54 is rational. accordingly. √3 √ 3 f. 1. Evaluate 64. f. 64 = 4 √ 3 2. The answer √ is a whole number, so 64 is rational. 3 classify 64. √ 3 √ 3 g. 1. Evaluate 32. g. 32 = 3.17480210394 … √ 3 2. The result is a non-terminating and 32 is irrational. √ 3 non-recurring decimal; classify 32. √ √ 3 1 3 1 1 h. 1. Evaluate. h. =. 27 27 3 √ 3 1 2. The result is a number in a rational form. is rational. 27 Resources Resourceseses 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). 1. WE1 √ 4 7 √ a. 4 b. c. d. 2 5 9 √ √ 1 √ 2. a. 7 b. 0.04 c. 2 d. 5 2 9 √ 3. a. b. 0.15 c. −2.4 d. 100 4 TOPIC 1 Indices, surds and logarithms 7  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 8 — #8 √ √ √ 25 4. a. 14.4 b. 1.44 c. 𝜋 d. 9 √ √ 5. a. 7.32 b. − 21 c. 1000 d. 7.216 349 157 … √ √ √ 3 1 6. a. − 81 b. 3𝜋 c. 62 d. 16 For questions 7 to 12, classify the following numbers as rational (Q), irrational (I) or neither. 1 √ 11 0 7. a. b. 625 c. d. 8 4 8 √ 1 √3 √ 1.44 8. a. −6 b. 81 c. − 11 d. 7 4 √ 8 √ 3 𝜋 9. a. 𝜋 b. c. 21 d. 0 7 √ √ 3 2 3 1 64 10. a. (−5) b. − c. d. 11 100 16 √ √ 2 6 √ 3 1 11. a. b. c. 27 d. √ 25 2 4 22𝜋 √ 3 √ √ 12. a. b. −1.728 c. 6 4 d. 4 6 7 13. MC Identify a rational number from the following. √ √ 4 9 √ 3 √ A. 𝜋 B. C. D. 3 E. 5 9 12 14. MCIdentify which of the following best represents an irrational number from the following numbers. √ 6 √3 √ √ A. − 81 B. C. 343 D. 22 E. 144 5 √ 𝜋 √ 15. MC Select which one of the following statements regarding the numbers −0.69, 7, , 49 is correct. 3 𝜋 A. is the only rational number. √3 √ B. 7 and 49 √ are both irrational numbers. C. −0.69 and 49 are the only rational numbers. D. −0.69 is the only rational number. √ E. 7 is the only rational number. 1 11 √ √ 3 16. MC Select which one of the following statements regarding the numbers 2 , − , 624, 99 is correct. 2 3 11 √ A. − and 624 are both irrational numbers. 3 √ √3 B. 624 is an irrational number and 99 is a rational number. √ √ 3 C. 624 and 99 are both irrational numbers. 1 11 D. 2 is a rational number and − is an irrational number. 2 3 √3 E. 99 is the only rational number. 8 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 9 — #9 Understanding √ a2 17. Simplify. b2 √ 18. MC If p < 0, then p is: A. positive B. negative C. rational D. irrational E. none of these √ 19. MC If p < 0, then p2 must be: A. positive B. negative C. rational D. irrational E. any of these Reasoning (√ √ ) (√ √ ) 20. Simplify p− q × p + q. Show full working. 21. Prove that if c2 = a2 + b2 , it does not follow that a = b + c. √ 22. Assuming that x is a rational number, for what values of k will the expression x2 + kx + 16 always be rational? Justify your response. Problem solving 36 23. Determine the value of m and n if is written as: 11 1 1 1 1 a. 3 + m b. 3 + c. 3 + d. 3 + n 3 + mn 3+ 1 m 1 3 + 1+ m n n −1 −1 1 3 −4 24. If x−1 means , determine the value of −1. x 3 + 4−1 1 3−n − 4−n 25. If x−n = n , evaluate −n when n = 3. x 3 + 4−n 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 eles-4662 A surd is an irrational number that is represented by a root sign or a radical sign, for example: √ √ 3 √ 4 , ,. √ √ √ 3 √4 Examples of surds include: 7, 5, 11, 15. √ √ √ √ The numbers 9, 16, 3 125, and 4 81 are not surds as they can be simplified to rational numbers, √ √ √3 √4 that is: 9 = 3, 16 = 4, 125 = 5, 81 = 3. TOPIC 1 Indices, surds and logarithms 9  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 10 — #10 WORKED EXAMPLE 2 Identifying surds Determine which of the following numbers √ are surds. √ √ 1 √ 3 √ 4 √ 3 a. 16 b. 13 c. d. 17 e. 63 f. 1728 16 THINK WRITE √ √ a. 1. Evaluate 16. a. 16 = 4 √ 2. The answer is rational (since it is a whole 16 is not a surd. number), so state your conclusion. √ √ b. 1. Evaluate 13. b. 13 = 3.605 551 275 46 … √ 2. The answer is irrational (since it is a 13 is a surd. non-recurring and non-terminating decimal), so state your conclusion. √ √ 1 1 1 c. 1. Evaluate. c. = 16 16 4 √ 1 2. The answer is rational (a fraction); state is not a surd. your conclusion. 16 √3 √ 3 d. 1. Evaluate 17. d. 17 = 2.571 281 590 66 … √ 3 2. The answer is irrational (a non-terminating 17 is a surd. and non-recurring decimal), so state your conclusion. √ 4 √ 4 e. 1. Evaluate 63. e. 63 = 2.817 313 247 26 … √ 4 √ 4 2. The answer is irrational, so classify 63 63 is a surd. accordingly. √ 3 √ 3 f. 1. Evaluate 1728. f. 1728 = 12 √ 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 eles-4663 In Mathematics you are required to study a variety of 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. a An irrational number is one that cannot be expressed in the form (where a and b are integers). The next √ b worked example sets out to prove that 2 is irrational. 10 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2022/4/28 — 19:45 — page 11 — #11 √ 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 = , where a and b are integers that have no a b written as in simplest form. We need to common factors and b ≠ 0. b show that a and b have no common factors. a2 2. Square both sides of the equation. 2= b2 3. Rearrange the equation to make a2 the subject a2 = 2b2 of the formula. 4. 2b2 is an even number and 2b2 = a2. ∴ a2 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 6. Square both sides. a2 = 4r2 But a2 = 2b2 from 7. Equate and. ∴ 2b2 = 4r2 4r2 b2 = 2 = 2r2 2 ∴ b is an even number and b must also be even; that is, b has a factor of 2. √ a 8. Use reasoning to deduce that 2= where Both a and b have a common factor of 2.√This b a a and b have no common factor. contradicts the original assumption that 2 = , b where √ a and b have no common factors. ∴ 2 is not rational. ∴ It must be irrational. 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  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 12 — #12 Exercise 1.3 Surds (10A) Individual pathways PRACTISE CONSOLIDATE MASTER 1, 4, 7, 8, 11, 14, 17 2, 5, 9, 12, 15, 18 3, 6, 10, 13, 16, 19 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. √ √ √ √ 1. a. 81 b. 48 c. 16 d. 1.6 √ √ √ √ 3 3 3 2. a. 0.16 b. 11 c. d. 4 27 √ √ √ √ 3. a. 1000 b. 1.44 c. 4 100 d. 2 + 10 √ 3 √ √ 3 √ 3 4. a. 32 b. 361 c. 100 d. 125 √ √ √ √ 3 7 5. a. 6+ 6 b. 2𝜋 c. 169 d. 8 √ (√ )2 √ √ 4 3 6. a. 16 b. 7 c. 33 d. 0.0001 √ 5 √ e. 32 f. 80 √ 6 √ √ √ 3 √ 7. MC The correct statement regarding the set of numbers { , 20, 54, 27, 9} is: 9 √ 3 √ A. 27 and 9 are the only rational numbers of the set. √ 6 B. is the only surd of the set. 9 √ 6 √ C. and 20 are the only surds of the set. 9 √ √ D. 20 and 54 are the only surds of the set. √ √ E. 9 and 20 are the only surds of the set. √ √ √ 1 3 1 1 √ √ 3 8. MC Identify the numbers from the set { , , , 21, 8} that are surds. 4 27 8 √ √ 1 A. 21 only B. only 8 √ √ 1 √ 3 1 √ C. and 8 D. and 21 only 8 8 √ 1 √ E. and 21 only 4 12 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/12/13 — 8:55 — page 13 — #13 √ 1 √ √ √ 9. MC Select a statement regarding the set of numbers {𝜋, , 12, 16, 3, +1} that is not true. 49 √ √ √ A. 12 is a surd. B. 12 and 16 are surds. √ √ C. 𝜋 is irrational but not a surd. D. 12 and 3 + 1 are not rational. E. 𝜋 is not a surd. √ √ 144 √ √ √ √ 10. MC Select a statement regarding the set of numbers {6 7, , 7 6, 9 2, 18, 25} that is not true. 16 √ √ 144 144 √ A. when simplified is an integer. B. and 25 are not surds. 16 16 √ √ √ √ C. 7 6 is smaller than 9 2. D. 9 2 is smaller than 6 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). √3 12. Determine the smallest value of m, where m is a positive integer, so that 16m is not a surd. Determine any combination of m and n, where m and n are positive integers with m < n, so that 13. a. √ 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. 5+ 2 b. 5− 2 c. 5+ 2 5− 2 15. WE3 Prove that the following numbers are irrational, using a proof by contradiction: √ √ √ a. 3 b. 5 c. 7. √ ( √ )( √ ) 16. 𝜋 is an irrational number and so is 3. Therefore, determine whether 𝜋 − 3 𝜋 + 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  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 14 — #14 1.4 Operations with surds (10A) 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 eles-4664 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, …). 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. The example below shows the how the surd 45 can be simplified by following the steps 1 to 3. √ √ 45 = 9 × 5 (Step 1) √ √ = 9× 5 (Step 2) √ √ = 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  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 15 — #15 Simplification of surds √ √ n = a2 × b √ √ = a2 × b √ = a× b √ =a b WORKED EXAMPLE 4 Simplifying surds Simplify the following surds. Assume that x and y are positive real numbers. √ √ 1√ √ a. 384 b. 3 405 c. − 175 d. 5 180x3 y5 8 THINK WRITE √ √ a. 1. Express 384 as a product of two factors where one a. 384 = 64 × 6 factor is the largest possible perfect square. √ √ √ 2. Express 64 × 6 as the product of two surds. = 64 × 6 √ 3. Simplify the square root from the perfect √ =8 6 square (that is, 64 = 8). √ √ b. 1. Express 405 as a product of two factors, one b. 3 405 = 3 81 × 5 of which is the largest possible perfect square. √ √ √ 2. Express 81 × 5 as a product of two surds. = 3 81 × 5 √ √ 3. Simplify 81. = 3×9 5 √ 4. Multiply together the whole numbers outside = 27 5 the square root sign (3 and 9). 1√ 1√ c. 1. Express 175 as a product of two factors c. − 175 = − 25 × 7 in which one factor is the largest possible 8 8 perfect square. √ 1 √ √ 2. Express 25 × 7 as a product of 2 surds. = − × 25 × 7 8 √ 1 √ 3. Simplify 25. = − ×5 7 8 5√ 4. Multiply together the numbers outside the =− 7 square root sign. 8 √ √ d. 1. Express each of 180, x3 and y5 as a product d. 5 180x3 y5 = 5 36 × 5 × x2 × x × y4 × y of two factors where one factor is the largest possible perfect square. √ √ 2. Separate all perfect squares into one surd and = 5× 36x2 y4 × 5xy all other factors into the other surd. TOPIC 1 Indices, surds and logarithms 15  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 16 — #16 √ √ 3. Simplify 36x2 y4. = 5 × 6 × x × y2 × 5xy √ 4. Multiply together the numbers and the = 30xy2 5xy pronumerals outside the square root sign. WORKED EXAMPLE 5 Multiplying surds Multiply the following surds, expressing answers in the simplest form. Assume that x and y are positive √ real √ numbers. √ √ √ √ √ √ a. 11 × 7 b. 5 3 × 8 5 c. 6 12 × 2 6 d. 15x5 y2 × 12x2 y THINK WRITE √ √ √ a. Multiply √ the √surds together, using a. 11 × 7 = 11 × 7 √ √ a × b = ab (that is, multiply = 77 expressions under the square root sign). Note: This expression cannot be simplified any further. √ √ √ √ b. Multiply the coefficients together and then b. 5 3×8 5 = 5×8× 3× 5 √ multiply the surds together. = 40 × 3 × 5 √ = 40 15 √ √ √ √ √ c. 1. Simplify 12. c. 6 12 × 2 6 = 6 4 × 3 × 2 6 √ √ = 6×2 3×2 6 √ √ = 12 3 × 2 6 √ 2. Multiply the coefficients together and = 24 18 multiply the surds together. √ 3. Simplify the surd. = 24 9 × 2 √ = 24 × 3 2 √ = 72 2 √ √ d. 1. Simplify each of the surds. d. 15x5 y2 × 12x2 y √ √ = 15 × x4 × x × y2 × 4 × 3 × x2 × y √ √ = x2 × y × 15 × x × 2 × x × 3 × y √ √ = x2 y 15x × 2x 3y √ 2. Multiply the coefficients together and the = x2 y × 2x 15x × 3y surds together. √ = 2x3 y 45xy √ = 2x3 y 9 × 5xy √ 3. Simplify the surd. = 2x3 y × 3 5xy √ = 6x3 y 5xy 16 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 17 — #17 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. (√ )2 ( √ )2 a. 6 b. 3 5 THINK WRITE (√ )2 (√ )2 a. Use a = a, where a = 6. a. 6 =6 (√ )2 √ ( √ )2 (√ )2 b. 1. Square 3 and apply a = a to square 5. b. 3 5 = 32 × 5 = 9×5 2. Simplify. = 45 1.4.2 Addition and subtraction of surds eles-4665 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 subtraction can take place. The concept of adding and subtracting surds is similar to adding and subtracting like terms in algebra. TOPIC 1 Indices, surds and logarithms 17  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 18 — #18 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. 3 6 + 17 6 − 2 6 b. 5 3 + 2 12 − 5 2 + 3 8 1√ √ √ c. 100a3 b2 + ab 36a − 5 4a2 b 2 THINK WRITE √ √ √ √ a. All 3 terms are √ alike because they contain a. 3 6 + 17 6 − 2 6 = (3 + 17 − 2) 6 √ the same surd ( 6). Simplify. = 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. Add like terms to obtain the simplified = 5 3+4 3−5 2+6 2 √ √ answer. = 9 3+ 2 1√ √ √ c. 1. Simplify surds where possible. c. 100a3 b2 + ab 36a − 5 4a2 b 2 √ √ 1 √ = × 10 a2 × a × b2 + ab × 6 a − 5 × 2 × a b 2 1 √ √ √ = × 10 × a × b a + ab × 6 a − 5 × 2 × a b 2 √ √ √ 2. Add like terms to obtain the simplified = 5ab a + 6ab a − 10a b √ √ answer. = 11ab a − 10a b TI | THINK DISPLAY/WRITE CASIO | THINK DISPLAY/WRITE a–c. a–c. a–c. a–c. In a new document, on a On the Main screen, Calculator page, complete complete the entry lines the entry lines as: as: √ √ √ √ √ √ 3 6 + 17 6 − 2 6 3 6 + 17 6 − 2 6 √ √ √ √ √ √ 5 3 + 2 12 − 5 2 5 3 + 2 12 − 5 2 √ √ +3 8 +3 8 ( √ 1√ √ simplify 1 100a3 b2 + 100a3 b2 + ab 36a− 2 2√ √ √ √ √ 3 6 + 17 √ 5 4a2 b|a > 0 and b > 0 √ √ 6 − 2√6 = 18√ 6 a × b × 36a− 5 3√+ 2 √12 − 5 2 + 3 8 √ Press ENTER after each 5 4a2 b|a > 0|b > 0) =9 3+ 2 entry. 1√ √ Press EXE after each √ √ √ √ 100a3 b2 + ab 36a− entry. 3 6 + 17√ 6 − 2√6 = 18√ 6 √ 2√ 2 √ 5 3√+ 2 √12 − 5 2 + 3 8 5 4a2 b = 11a 3 b − 10a b =9 3+ 2 1√ √ 100a3 b2 + ab 36a− 2√ 2 √ 5 4a2 b = 11a 3 b − 10a b 18 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 19 — #19 1.4.3 Dividing surds eles-4666 To divide surds, divide the expressions under the radical signs. Dividing surds √ √ a a √ = b 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 48 9 88 36xy a. √ b. √ c. √ d. √ 5 3 6 99 25x9 y11 THINK WRITE √ √ √ √ a a 55 55 a. 1. Rewrite the fraction, using √ =. a. √ = b b 5 5 √ 2. Divide the numerator by the denominator = 11 (that is, 55 by 5). Check if the surd can be simplified any further. √ √ √ √ a a 48 48 b. 1. Rewrite the fraction, using √ =. b. √ = b b 3 3 √ 2. Divide 48 by 3. = 16 √ 3. Evaluate 16. =4 √ √ √ √ a a 9 88 9 88 c. 1. Rewrite surds, using √ =. c. √ = b b 6 99 6 99 √ 9 8 2. Simplify the fraction under the radical by = dividing both numerator and denominator 6 9 by 11. TOPIC 1 Indices, surds and logarithms 19  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 20 — #20 √ 9×2 2 3. Simplify surds. = 6×3 √ 18 2 4. Multiply the whole numbers in the numerator = together and those in the denominator 18 together. √ 5. Cancel the common factor of 18. = 2 √ √ 36xy 6 xy d. 1. Simplify each surd. d. √ = √ 25x9 y11 5 x8 × x × y10 × y √ 6 xy = 4 5√ 5x y xy 6 2. Cancel any common factors — in this √ = case xy. 5x4 y5 1.4.4 Rationalising denominators eles-4667 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. 6 2 12 a. √ b. √ 13 3 54 √ √ 17 − 3 14 c. √ 7 20 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 21 — #21 THINK WRITE √ 6 a. 1. Write the fraction. a. √ 13 √ √ 6 13 2. Multiply both the numerator and denominator by the √ =√ ×√ 13 13 surd contained in the denominator (in this case 13). √ This has the same √ effect as multiplying the fraction 78 = 13 13 by 1, because √ = 1. 13 √ 2 12 b. 1. Write the fraction. b. √ 3 54 √ √ 2 12 2 4 × 3 2. Simplify the surds. (This avoids dealing with large √ = √ numbers.) 3 54 3 9 × 6 √ 2×2 3 = √ 3×3 6 √ 4 3 = √ 9 6 √ √ √ 4 3 6 3. Multiply both the numerator and denominator by 6. = √ ×√ This has √ the same effect as multiplying the fraction by 1, 9 6 6 √ 6 4 18 because √ = 1. = 6 9×6 Note: We need to multiply√ only by the surd part √ of the denominator (that is, by 6 rather than by 9 6.) √ √ 4 9×2 4. Simplify 18. = 9×6 √ 4×3 2 = 54 √ 12 2 = 54 √ 2 2 5. Divide both the numerator and denominator by 6 = (cancel down). 9 √ √ 17 − 3 14 c. 1. Write the fraction. c. √ 7 √ √ √ √ ( 17 − 3 14) 7 2. Multiply both the numerator and denominator by 7. = √ ×√ Use grouping symbols (brackets) to make it √ clear that 7 7 the whole numerator must be multiplied by 7. TOPIC 1 Indices, surds and logarithms 21  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 22 — #22 √ √ √ √ 3. Apply the Distributive Law in the numerator. 17 × 7 − 3 14 × 7 a (b + c) = ab + ac = √ √ 7× 7 √ √ 119 − 3 98 = 7 √ √ √ 119 − 3 49 × 2 4. Simplify 98. = √ 7 √ 119 − 3 × 7 2 = √ 7 √ 119 − 21 2 = 7 1.4.5 Rationalising denominators using conjugate surds eles-4668 The product of pairs of conjugate surds results in a rational number. √ √ √ √ √ √ Examples of pairs of conjugate surds include 6 + 11 and 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. Using conjugates to rationalise the denominator To rationalise the denominator that contains a sum or a difference of surds, multiply both numerator and denominator by the conjugate of the denominator. Two examples are given below: √ √ 1 b a− 1. To rationalise the denominator of the fraction √ √ , multiply it by √ √. a+ b a− b √ √ 1 a+ b 2. To rationalise the denominator of the fraction √ √ , multiply it by √ √. a− b a+ b A quick way to simplify the denominator is to use the difference of two squares identity: (√ √ ) (√ √ ) (√ )2 (√ )2 a− b a+ b = a − b = a−b WORKED EXAMPLE 10 Using conjugates to rationalise the denominator Rationalise the denominator and simplify the following. 1 a. √ 4− 3 √ √ 6+3 2 b. √ 3+ 3 22 Jacaranda Maths Quest 10 + 10A  “c01IndicesSurdsAndLogarithms_PrintPDF” — 2021/9/8 — 12:27 — page 23 — #23 THINK WRITE 1 a. 1. Write the fraction. a. √ 4− 3 √ 1 (4 + 3) 2. Multiply the numerator and = √ × √ denominator by the conjugate of the (4 − 3) (4 + 3) denominator. √ (4 + 3) (Note that √ = 1). (4 + 3) √ 4+ 3 3. Apply the Distributive Law in the = √ 2 numerator and the difference of two (4)2 − ( 3) squares identity in the denominator. √ 4+ 3 4. Simplify. = 16 − √3 4+ 3 = 13 √ √ 6+3 2 b. 1. Write the fraction. b. √ 3+ 3 √ √ √ ( 6 + 3 2) (3 − 3) 2. Multiply the numerator and = √ × √ denominator by the conjugate of the (3 + 3) (3 − 3) denominator. √ (3 − 3) (Note that √ = 1.) (3 − 3) √ √ √ √ √ √ 3. Multiply the expressions in grouping 6 × 3 + 6 × (− 3) + 3 2 × 3 + 3 2 × −( 3) symbols in the numerator, and apply the = √ 2 difference of two squares identity in the (3)2 − ( 3) denominator. √ √ √ √ 3 6 − 18 + 9 2 − 3 6 4