Year 07 - Mathematics - Unit 1-4

Year 07 - Mathematics - Unit 1-4

Mathematics (9-1) - iGCSE 2018-20 Year 07 Unit 1 Number Contents 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Number Prior knowledge check Number problems and reasoning Place value and estimating HCF and LCM Calculating with powers (indices) Zero, negative and fractional indices Powers of 10 and standard form Surds Problem-solving Checkup Strengthen Extend Knowledge check 7

7 -- Surds Surds Page Page iv iv 1 1 2 4 6 8 11 14 16 19 20 22 25 27 Contents 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 At the end of the Master lessons, take a check-up test to help you to decide whether to strengthen or Extend our learning Unit Openers put the maths you are about to learn into a real-life context. Have a go at the

question - it uses maths you have already learnt so you should be able to answer it at the start of the unit. Use the Prior knowledge check to make sure you are ready to start the main lessons in the unit. It checks your knowledge from Key Stage 3 and from earlier in the GCSE course. Your teacher has access to worksheets if you need to recap anything. 7 7 -- Surds Surds Page Page vv Extend helps you to apply the maths you know to some different situations When you have finished the whole unit, a Unit test helps you see how much progress you are making. Choose only the topics in strengthen thay you need a bit more practice with. Youll find more hints here to lead you through specific questions. Then move on to Extend Contents 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5

Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page vi vi Contents 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page vi vi 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value

Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Warm Up Numerical fluency 1. Work out a. 5 x 0.3 b. 97 x 0.02 c. 6 0.2 d . 27 0.09 e. 4.2 0.1 f . 0.4 x 0.6 g. 0.9 x 0.02 h . 0.09 x 0.09 i. 0.4 0.2 j . 0.9 0.03 k. 0.45 0.3 l . 0.88 0.04 2. Choose the correct sign, < or >. a. 2.7 2.5 b. 3.04 3.3 c. -2.9 -2.8 d. -5.16

7 7 -- Surds Surds Page Page 138 138 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 4. a. Copy and complete the Venn diagram to show the factors of 16 and 20. b. Write down the highest common factor. 5. a Write down the first 10 multiples of 6 and 9. b. Make a list of the common multiples. c. Write down the lowest common multiple. 7 7 -- Surds Surds Page Page 1 1 1.1 Number Problems and Reasoning 1

1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. a. Copy and complete the Venn diagram to show the first 10 multiples of 4 and 10. b. Write down the lowest common multiple of 4 and 10. 7. Work out a. 8 2 x 3 b. (8 - 2) x3 c. 7 - (4 - 1) x 6 d. 24 (8 - 2) e. 42 +1 f. (-6)2 8. Insert brackets to make this calculation correct. 7 7 -- Surds Surds Page Page 1 1 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value

Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 9. Estimate a. 7.3 x 8.94 b. 47 2.1 b. 5.2 + 4.9 d. 7.9 2.4 10. Write the positive and negative square roots of these numbers. a. 36 b. 1 c. 64 11. Work out a. 4 x 9 x 25 b. 102 x 48 b. 182 x 99 d. 27 x 6 + 27 x 4 12. Copy and complete. a. 6 x 6=6 b. 3 x 3 x 3 7 7 -- Surds Surds Page Page 1 1 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2

Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 * Challenge 14. How many different ways can these cards be arranged? What about now? 7 7 -- Surds Surds Page Page 2 2 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 7 7 -- Surds Surds Page Page 2 2 Objectives Why Learn This Work out the total number of ways of performing a series of tasks. 5! in maths means five factorial and is equal to 5 x 4 x 3 x 2 x 1. Fluency Work out 4x4x4 10 x 9 x 8 5x4x3 4x3x2x1 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 1. a. Copy and complete this list of all possible outcomes for rolling a dice and flipping a coin. H, 1 H, 2 T, 1 ................... b. How many outcomes are there altogether? 2. a. Copy and complete this list of all possible outcomes for spinner A and spinner B. 2, 1 4, 1 6,1 2, 3 4, 3 .. 2, 5 .. .. b. How many outcomes are there altogether. 7 7 -- Surds Surds Page Page 2 2 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers

Powers of of 10 10 3. How many possible outcomes are there when a rolling a. dice b. flipping a coin c. spinning A in Q2 d. spinning B in Q2? Discussion What do you notice about your answers to i. Q1b and Q3a and b above ii. Q2b and Q3c and d above? ActiveLearn - Homework, practice and support: Higher 1.1 7 7 -- Surds Surds Page Page 2 2 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 4. 7 7 -- Surds Surds

Page Page 3 3 A restaurant offers a set menu for birthday parties, a. Write down all possible combinations of starters and main courses. b. Reflect How did you order your list to make sure Starters Vegetable Soup (v) you didnt miss any starters or mains? Salad (S) The restaurant decides to offer fish (F) as a main course Melon (M) c. How many possible combinations are there now? Mains d. Copy and complete. Pizza (P) Spaghetti 3 starters and 4 mains: combinations Bolognaise (B) 3 starters and 5 mains: combinations Curry (C) n starters and m mains: combinations Lasagna (L) Another caf offers 2 starters, 4 mains and 3 desserts. e. How many possible combinations are there now? Q4a Q4a hint hint -- Use Use letters letters for for combinations, combinations, for for example example VP VP for for vegetable vegetable soup, soup, pizza. pizza. Key Point 1 When When there there are are m m ways ways of of doing doing one one task task and and nn ways

ways of of doing doing aa second second task, task, the the total total number number of of ways ways of of doing doing the the first first task task then then the the second second task task is is m m xx n. n. 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5 Exam-Style Questions Jess has a 4-digit password for her mobile phone. Each digit can be between 0 and 9 inclusive. a. How many choices are possible for each digit of the code?

b. What is the total number of 4-digit passwords that Jess can create? Jess would like to choose an even number. The code can start with a zero. c. How many different ways are possible now? Q5 hint Q5 communication communication hint -- Inclusive Inclusive (5 marks) means means that that the the end end numbers numbers are are also also included. included. 7 7 -- Surds Surds Page Page 3 3 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 6. Three people, A, B and C, enter a race. a. Write down the different orders in which they can finish first, second and third. Harry says that there are 3 possible winners, but then only 2 possibilities for second place and only one person left for third place. Discussion Is Harry correct? Explain your answer. b. How many different ways can people finish in i. a 4-person race ii a 6person race iii. a 10-person race? Q6 communication Q6 communication hint hint -- A A factorial factorial is is the the result result of of multiplying multiplying aa sequence sequence of of descending descending integers. integers. For For example example '4 '4 factorial' factorial' == 4! 4! == 44 xx 33 xx 22 xx 1. 1. Make Make sure sure you you know know how how to to use use the the factorial factorial button button on on your your calculator. calculator.

7 7 -- Surds Surds Page Page 3 3 1.1 Number Problems and Reasoning 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Problem-solving Eddie needs to choose a 6-digit code for his computer password, a. How many codes can Eddie create using i. 6 numbers ii. 4 numbers followed by 2 letters iii. 1 number followed by 5 letters? Eddie decides that he does not want to repeat a digit or a letter, b. How many ways are possible in parts i to iii now? 7 7 -- Surds Surds Page Page 3 3

1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Objectives 7 7 -- Surds Surds Page Page 4 4 Why Learn This Builders use Builders use Estimate an answer. estimates to give their Estimate an answer. estimates to give their Use place value to clients an idea of how Use place value to clients an idea of how answer questions. much the work will answer questions. much the work will cost. cost. Fluency

Fluency Which two whole numbers does each square root lie between? 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Warm Up 1. Write each number to i. 1 significant figure ii. 2 significant figures. a. 873 209 b. 2019 c. 0.007 059 2. Work out a. 9 x (4 + 7) b. 5 + 3 x 8 c. 7 x 5 4 x 2 d. 30 5 x 8 e. 72 9 f. 3. Work out the mean of 3, 6, 7, 9, 15 and 20. 4. Work out a. 32 x 6 b. 16 x 12 c. 8 x 24 d. 4 x 48 7 7 -- Surds Surds

Page Page 4 4 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. 3.7 x 9.86 = 36.482 Use this fact to work out the calculations below. Check your answers using an approximate calculation. a. 37 x 9.86 b. 3.7 x 9.86 Q5d Q5d hint hint -c. 0.0037 x 98.6 Rewrite the the given given d. 36.482 9.86 Rewrite calculation calculation as as aa e. 3648.2 98.6 division. division. f. 364.82 370 Q5a hint Compare with the given calculation. X X 7

7 -- Surds Surds Page Page 4 4 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. Reasoning 54.8 x 7.29 = 399.492 a. Write down three more calculations that have the same answer, b. Write down a division that has an answer of 54.8. c. Write down a division that has an answer of 0.729. d. Charlie says that 54.8 x 72.9 = 3989.44 Explain why Charlie must be wrong, 7 7 -- Surds Surds Page Page 4 4 1.2 Place Value and Estimating 1

1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. a Write down the value of 74 and 79. b. Estimate the value of 75, 76, 77 and 78. Round each estimate to 1 decimal place c. Use a calculator to check your answer to part b. Q7b hint Use a number line to help. 7 7 -- Surds Surds Page Page 4 4 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers

Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. a Write down the value of 74 and 79. b. Estimate the value of 75, 76, 77 and 78. Round each estimate to 1 decimal place c. Use a calculator to check your answer to part b. 8. Estimate the value to the nearest tenth, a. b. c b. e. f. Q8a hint You can write your answer means 'is approximately equal to' 7 7 -- Surds Surds Page Page 4 4 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5

Indices Indices 6 6 Powers Powers of of 10 10 ActiveLearn - Homework, practice and support: Higher 1.2 9. Problem-solving A mosaic uses 150 square tiles. The total area is 3000cm2. a. Estimate the side length of a tile. b. Use a calculator to check your answer. 10. a. Write down the value of 82 and 92. b. Estimate the value of 8.32 and 8.82. Round each estimate to the nearest whole number. c. Use a calculator to check your answer to part b. 7 7 -- Surds Surds Page Page 5 5 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 11. Estimate to the nearest whole number. a. 3.22 b. 4.72 c. 1.72 b. 7.12 e. 6.32 f. 9.82 Q11a hint - Use a number line to help. 322 3.222 422 Q13 hint Work out the 2 value of the 5 + 80 13. The sum of the values card on the on 42 these cards 2 x is4 12. right first. 7 7 -- Surds Surds Page Page 5 5 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 12. a. 7 7 -- Surds Surds Page Page 5 5 Estimate answers to these. i. (11.2 - x 4.08 ii. (1.98 x 3.14)2 8.85 iii. b. Use your calculator to work out each answer. Give your answers correct to one decimal place. c. Reflect How did you decide what to round each number to? For iii and iv does it matter if you round the numerator or the denominator first? Q12a hint - The whole of the expression is being square rooted. So estimate the numerator and denominator before square square rooting. rooting. 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 14. Problem-solving A large dice has a side length of 9.2 cm. Estimate the surface area of the cube. 15. Problem-solving The area of a square is 80cm2. Estimate the perimeter of the square. 16. Problem-solving Pieces of turf are 1 m long by 0.5m wide. Each bit costs 3.79. a. Estimate the cost of turf required to cover these spaces. i. 9.6 m by 2.4 m ii. 6.2 m by 1.9 m iii. 4.4 m by 2.1m b. Use a calculator to work out each answer. How good were your estimates? Discussion Is it better to overestimate or underestimate a cost? 7 7 -- Surds Surds Page Page 5 5 1.2 Place Value and Estimating 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

7 7 -- Surds Surds Page Page 5 5 17. STEM Robert uses a spreadsheet to record his runs for 10 innings. His scores are in cells A1 to J1. His mean score is in cell K1. a. Use estimates to show that Robert's mean is wrong, b. Work out Robert's correct mean to the nearest tenth. A B C D E F G H I J K 1 76 12 4 15 0 35 0 7 12 21 11.8 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Objectives Write a number as the product of its prime factors.

Find the LCF and LCM of two numbers Fluency 7 7 -- Surds Surds Page Page 6 6 Why Learn This Astronomers use the lowest common multiple of patterns in the orbits of the Sun and Moon to predict solar eclipses. Work out 2 x 3 x 52 23 x 32 5 x 5 x 5 = 5 7 x 7 x 7 x 7 x 7 x 7 = 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Warm Up 1. a. Write down all the factors of 20. b. Which of these factors are prime numbers? 2. a.

Write down all the prime numbers between 1 and 20. b. Write down all the factors of 24. c. Copy and complete this Venn diagram. 7 7 -- Surds Surds Page Page 6 6 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3. a. Copy and complete this factor tree for 40. b. Write 40 as a product of its prime factors. Q3b hint - Circle the prime factors in your factor tree. 4. Write 75 as a product of its prime factors. Q4 hint - Use the method from Q3.

7 7 -- Surds Surds Page Page 6 6 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Steve and Ian are asked to find 60 as a product of its prime factors. Steve begins by writing 60 = 5 x 12 Ian begins by writing 60 = 6 x 10 a. Work out a final answer for Steve. b. Work out a final answer for Ian. Discussion What do you notice about the two answers? c. Start the prime factor decomposition of 48 in two different ways: 6x8 and 12 x 4. Discussion Does your first step in a prime factor decomposition affect your final answer? 7 7 -- Surds Surds Page Page 6 6

1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. Reflect a. Write down your own short mathematical definition of these words, i. prime ii factor iii. decomposition b. Use your definition to write down (in your own words) the meaning of prime factor decomposition. ActiveLearn - Homework, practice and support: Higher 1.3 7 7 -- Surds Surds Page Page 6 6 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value

Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Write each number as a product of its prime factors in index form, a. 18 b. 42 c. 25 d. 36 e. 24 f. 80 Q7 communication hint - In index form means to write a number to a power or an index. 233 is written in index form. 3 is the power or index. 8. 120 can be written as a product of its prime factors in the form 2m x n x p. Work out m, n and p. 7 7 -- Surds Surds Page Page 7 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF

HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 7 7 Example 1 Find the highest common factor and lowest common multiple of 24 and 60. 24 = 2 x 2 x 2 x 3 Write each number as a 60 = 2 x 2 x 3 x 5 product of prime factors Draw a Venn Diagram The highest common factor (HCF) of 24 and 60 Multiply the common prime factors = 2 x 2 x 3 = 12 The lowest common multiple (LCM) of 24 and 60 Multiply all the prime factors = 2 x 2 x 2 x 3 x 5 = 120 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5

5 Indices Indices 6 6 Powers Powers of of 10 10 9. Find the HCF and LCM of a. 24 and 30 b. 20 and 42 c. 8 and 18 d. 15 and 45 Q9 hint Draw a Venn diagram for e. 27 and 36 f. 35 and each question to help you. 66 10. Real / Problem-solving One bus leaves the bus station every 15 minutes. Another bus leaves every 12 minutes. At 2:30 pm both buses leave the 7 7 -- Surds Surds Page Page 7 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6

6 Powers Powers of of 10 10 11. Real / Problem-solving Amber wants to tile her bathroom. It measures 1.2 m by 2.16 m. She finds square tiles with a side length of 10cm, 12cm or 18 cm. Which of these tiles will fit the wall exactly? Discussion How do you know whether to find the HCF or LCM for Q10 and Q11? 12. Problem-solving The HCF of two numbers is 2. Write down three possible pairs of numbers. Q12 hint First choose two numbers where 2 is a factor. Is 2 the highest common factor of these numbers? 7 7 -- Surds Surds Page Page 7 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

13. Problem-solving The LCM of two numbers is 18. One of the numbers is 18. a. Write down all the possibilities for the other number. b. Describe the set of numbers you have created. 14. 48 = 24 x 3 and 36 = 22 x 32 Write down, as a product of its prime factors, a. the HCF of 48 and 36 b. the LCM of 48 and 36. Q14 hint You could draw a Venn diagram. 7 7 -- Surds Surds Page Page 7 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 15 Exam-Style Questions Given that A = 23 x 34 x 52 and B = 22 x 36 x 5 Write down, as a product of its prime factors, a. the HCF of A and B

b. the LCM of A and B. Exam hint (2as marks) 'Write a product of its prime factors' means you dont have to calculate the number. 7 7 -- Surds Surds Page Page 8 8 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 16. Write 80 as a product of its prime factors. Discussion How can you use the prime factor decomposition of 80 to quickly work out the prime factor decomposition of 160? What about 40? 17. Problem-solving The prime factor decomposition of 2100 is 22 x 3 x 52 x 7. Write down the prime factor decompositions of a. 75

b. 24 c. 12 d. 30 7 7 -- Surds Surds Page Page 8 8 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 18. a. Harry says the prime factors of 75 appear in the prime factor decomposition of 2100, so 2100 is divisible by 75. Is 2100 divisible by 24,12 or 30? b. Use prime factors to show that 792 is divisible by 12. c. Is 792 divisible by 132? Explain your answer, d. Is 792 divisible by 27? Explain your answer. 7 7 -- Surds Surds

Page Page 7 7 1.3 HCF and LCM 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 7 7 19. In prime factor form, 700 = 22 x 52 x 7 and 1960 = 23 x 5 x 72 a. What is the HCF of 700 and 1960? Give your answer in prime factor form. b. What is the LCM of 700 and 1960? Give your answer in prime factor form. c. Which of these are factors of 350 and 1960? i. 2 x 5 x 7 Q19c Q19c hint hint -- What What factors factors do do 700 700 and and ii. 49 1960 1960 have

have in in common? common? Any Any factor factor of of this this iii. 20 number number will will be be aa factor factor of of 350 350 and and 1960. 1960. 2 2 iv. 2 x 5 x 7 d. Which of these are multiples of 350 and 1960? i. 23 x 5 x 73 Q19d Q19d hint hint -- What What multiples multiples do do 700 700 and and ii. 26 x 52 x 72 1960 1960 have have in in common? common? Any Any multiple multiple of of this this iii. 22 x 5 x 7. number number will will be be aa multiple multiple of of 350 350 and and 1960. 1960. 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2

2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Objectives Use powers and roots in calculations. Multiply and divide using index laws. Work out a power raised to a power. Fluency Work out 62 (-4)2 24 7 7 -- Surds Surds Page Page 6 6 Why Learn This A googol is a 1 followed by 100 zeros. It can be written as 10100. 15 ActiveLearn - Homework, practice and support: Higher 1.4 1.4 Calculating With Powers (Indices) 1

1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 1. Work out a. 33 b. (-1)3 c. 4 x 42 d. 32 x 5 e. 23 x 102 f. 0.23 g. 3 x xx xx == 64 64 Warm Up h. x 2. Work out a. b. 3. Copy and complete. a. 2 =16 Q3a Q3a hint hint b. 3 = 64 22 xx 22 xx ... ... == 16 16 c. 5 = 25 d. 3 = 27 Q3b Q3b hint hint 7 7 -- Surds Surds

Page Page 9 9 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Key Point 1 The inverse of a cube is the cube root. 233 = 8, so the cube root of 8 is = 2 4. Work out a. b. c. Discussion Why it is possible to find the cube root of a negative number, but not the square root? 6. a. [(33 52) x 2)]3 b. 20 [3 x 42 (22 x 32)] c. [72 (7 5)3 3] Q6 Q6 communication communication hint hint -- Square Square brackets

brackets [[ ]] make make the the inner inner and and outer outer brackets brackets easier easier to to see. see. 7 7 -- Surds Surds Page Page 9 9 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Work out these. Use a calculator to check your answers. a. 43 - Q5a Q5a hint hint -- Use Use the the priority

priority of of operations. operations. b. 33 Q5e Q5e hint hint -- The The square square root root applies applies to to the the whole whole calculation. calculation. Work Work out out the the cube cube root root first. first. 7 7 -- Surds Surds Page Page 9 9 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6

6 Powers Powers of of 10 10 7. Work out a. Q7a hint - 44 = 16 b. c. = 8. a. Work out i. 103 x 102 ii 105 iii. 106 x 102 iv 108 b. How can you work out the answers to part a by using the indices of the powers you are multiplying? Q8c c. Check your rule works forii hint i. 103 x 104 10 = 1011 ii. 105 x 10 7 7 -- Surds Surds Page Page 9 9 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM

LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 9. Write each product as a single power. a. 32 x 34 b. 42 x 48 3 4 c. 9 x 9 Key Point 3 To multiply, add the indices xmm x xnn = xmm ++ nn 10.Find the value of a. a. 84 x 8a = 87 7 7 -- Surds Surds Page Page 9 9 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices

Indices 6 6 Powers Powers of of 10 10 11. Write these calculations as a single power. Give your answers in index form, a. 27 x 35 = 3 x 35 = 3 b. 43 x 64 c. 5 x 125 d. 32 x 4 e. 8 x 8 x 8 4 Key Point f. 9 x 27 x 3 You can only add the indices when multiplying powers of the same number. 14.Find the value of a. 6 a 4 5 7 7 -- Surds Surds Page Page 10 10 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5

Indices Indices 6 6 Powers Powers of of 10 10 12. Reasoning a. i Work out by cancelling. Write your answer as a power of 5. ii Copy and complete. 55 + 52 = 5 b. Copy and complete 46 x 42 = = 4 c. Work out 65 64 Discussion How can you quickly find 79 73 without writing all the 7s? 13. Workout a. 76 + 72 b. 45 43 c. 36 35 Key Point 5 To divide powers, subtract the indices. xmm xaa = 7mm -- aa 7 7 -- Surds Surds Page Page 10 10 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5

Indices Indices 6 6 Powers Powers of of 10 10 15. Problem-solving a. Yu multiplies three powers of 9 together. 9 x 9 x 9 = 912 What could the three powers be when i. all three powers are different ii. all three powers are the same? b. Harvey divides two powers of 5. 5 5 = 56 What could the two powers be when i. both numbers are greater than 520 ii. the power of one number is double the power of the other number? 7 7 -- Surds Surds Page Page 10 10 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 16. Work out these. Write each answer as a single power. a. 53 x 57 54 b. 63 62 x 67 c. d. 17. Real / STEM The hard drive of Tom's computer holds 23S bytes of data. He buys a USB memory stick that holds 236 bytes of data. a. How many memory sticks does he need to back up his computer? b. He buys an external hard drive that holds 239 bytes of data. c. What fraction of the external hard drive does he use when backing up his computer? 7 7 -- Surds Surds Page Page 10 10 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 18. Copy and complete. a. (23)5 = 23 x x x x = 2 b. (64)3 = x x = 6 c. (87)2 = x = 8 Discussion What do you notice about the powers in the question and the powers in the final answer? Key Point 6 To work out a power to another power, mn multiply the powers together. (xmm)nn = xmn 7 7 -- Surds Surds Page Page 11 11 1.4 Calculating With Powers (Indices) 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 19. Write as a single power. a. (23)4 b. (62)5 c. (42)-3 20. Problem-solving Write each calculation as a

single power. a. 8 x 32 x 8 b. Key Point 6 To work out a power to another power, mn multiply the powers together. (xmm)nn = xmn 7 7 -- Surds Surds Page Page 11 11 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Objectives Use negative Use negative indices. indices. Use fractional Use fractional indices. indices. Fluency Fluency 7 7 -- Surds

Surds Page Page 11 11 Why Learn This The smallest known time The smallest known time measurement is approximately measurement is approximately 1043 seconds. Scientists call 43 10 seconds. Scientists call this unit one Planck time, after this unit one Planck time, after Max Planck. Max Planck. Work out Convert 0.3 to a fraction 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 1. Work out a. 62

b. 23 c. 34 d. 53 2. Write each calculation as a single power, a. 34 x 36 b. 25 23 b. 16 x 8 d. 73 x 75 3. Work out a. b. c. d. 7 7 -- Surds Surds Page Page 11 11 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 11

11 4. Work out the value of n. Q4a hint - 40 = 5 x a. 40 = 5 x 2n b. 3n x 3n = 38 How do you write this c. 52n 5n = 56 d. x 4n = 32 number as 2? 5. a. Use a calculator to work out i. 2-1 ii. 4-1 iii. 5-1 iv. 10-1 b. Write your answers to part a as fractions, c. Use a calculator to work out i. 2-2 ii 4-2 iii. 5-2 iv 10-2 d. Write your answers to part c as fractions, e. Work out i. ()-1 ii. ()-2 Discussion What is the rule for writing negative indices as fractions? ActiveLearn - Homework, practice and support: Higher 1.5 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. a. Match the equivalent cards. b. Write a matching tile for the two tiles that are left over, c. Copy and complete ()-1 = , so ()-1 = ()

Key Point 7 x-a-a = for any number n, x 0 7 7 -- Surds Surds Page Page 12 12 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Work out these. Write each answer as a single power. a. 62 6-3 x 67 b. c. 8. Problem-solving a. Copy and complete. 23 x 23 = 2 b. Write down 23 as a whole number. c. 23 23 = 8 = d. Copy and complete using parts a and c. 23 23 = 2 = e. Repeat parts a and b for 75 75. f. Write down a rule for a6, where a is any number. Key Point 8 x00 = where x is any non-zero number. 7

7 -- Surds Surds Page Page 12 12 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 9. Work out a. 3-1 10-5 d. ()-1 b. 2-4 c. e. -3 f. (1)-1 g. (2)-2 h. (0.7)-1 (0.1)-5 (0.4)-3 Q9fi.strategy hint Convertj.mixed numbers to improper fractions Q9h strategy hint Convert decimals

to fractions k. (5-1)0 l. (7-1)-1 7 7 -- Surds Surds Page Page 12 12 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 10. a. Use a calculator to work out i. 49 ii. 16 iii. 121 iv. () b. Copy and complete. a is the same as the ______ _______ of a. c. Work out i. 271/3 ii. 10001/3 iii. -11/3 iv. () 1/3

d. Copy and complete. a1/3 is the same as the ______ _______ of a. e. Copy and complete. i. 625 = 5 so 6251/4 = 7 7 -- Surds Surds Page Page 12 12 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 11.Evaluate a. 361/2 b. 1/2 c. d. b. 811/2 d. ()1/2 f. -81/3 h. ()1/3 Q11 communication hint - Evaluate means work out the value of.

Key Point 9 1/n 1/n = 7 7 -- Surds Surds Page Page 12 12 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 12. Workout a. 25-1/2 b. 64-1/3 ()-1/2 -1/2 = = Q12 hint - x-n-n = so 25-1/2 c. Example 2 2/3 b. 16-3/4 -3/4 Work out the value of a. 272/3 a. b.

2/3 = (272/3 2/3)22 = 322 = 9 272/3 -3/4 = = = 16-3/4 mn. Use the rule (xmm)nn = xmn Work out the cube root of 27 first. Then square your answer. Use x-n-n = 7 7 -- Surds Surds Page Page 13 13 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 13. Work out a. 642/3 100003/4 c. 163/2 ()3/2 Q13 hint -2/3 e.

27 2/3 = (641/3 1/3)22 = (22 = 642/3 81-3/4 Key Point 10 n/m = ( xn/m b. d. f. - 7 7 -- Surds Surds Page Page 13 13 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 14. Work out a. 27-1/3 x 93/2 b. ()-3/2 x ()1/3 c. ()3/4 x ()-3/2 15. Find the value of n. a. 16 = 2n c. 3/4

d. n Q14a Q14a hint hint First First work work out out -1/3 . 27 27-1/3 . Then Then work work out out 3/2. 993/2 . The The multiply multiply these these numbers numbers together. together. b. x -3/2 e. 7 7 -- Surds Surds Page Page 13 13 1.5 Zero, Negative and Fractional Indices 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF &

& LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 16. Problem-solving / Reasoning Will says that 25- x 642/3 = 80 a. Show that Will is wrong, b. What mistake did he make? 17. Problem-solving / Reasoning Match the expressions with indices to their values. 7 7 -- Surds Surds Page Page 13 13 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds

Surds Page Page 14 14 Objectives Why Learn This Write a number in Write a number in Scientists use standard standard form Scientists use standard form to write very small or standard form Calculate with numbers form to write very small or Calculate with numbers very large numbers. in standard form. very large numbers. in standard form. Fluency Fluency Work out 845.3 x 0.001 Which of these are the same as 10? x x 0.01 x 10-1 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

Warm Up 1. Copy and complete. If your answer is a fraction, write it as a decimal too. a. 10 = b. 10-1 = c. 10-2 = d. 10-3 = e. 10-4 = f. 10-5 = 2. Write down the value of x. a. 10x = 1000 b. 105 = x c. 10x = 100 000 000 d. 10-1 = x e. 10x = 0.0001 f. 10-6 =x 3. Copy and complete a. 5 670 000 = million 7 7 -- Surds Surds Page Page 14 14 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

7 7 -- Surds Surds Page Page 14 14 4. Copy and complete the table of prefixes. Prefix Tera Giga Mega Kilo Deci Centi Milli Micro Nano pico Letter T G M k d c m n p Power 1012 109 Number 1 000 000 000 000 1 000 000 103 0.1 10-2 0.001 10-6 0.000 000 001 10-12 Q4 communication hint - Prefix is the beginning part of a word

Q4 communication hint - , the letter for the prefix micro, is the Greek letter for mu. 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Convert a. 15 mg into grams b. 7 nm into metres c. 1.7 g into kg d. 7.3 ps into seconds. Q5a hint Use a number line ActiveLearn - Homework, practice and support: Higher 1.6 7 7 -- Surds Surds Page Page 14 14 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2

Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. STEM Write these measurements in metres. a. The size of the influenza virus is about 1.2pm. b. The radius of a hydrogen atom is 25 pm. c. A fingernail grows about 0.9 nm every second. 7. Copy and complete. a. 45 000 = 4.5 x b. 10 000 = 10 c. 45 000 4.5 x 10 7 7 -- Surds Surds Page Page 14 14 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5

5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 15 15 Key Point 12 A number is in standard form when it is in the form A x 10nn, where 1 A < 10 and n is an integer. For example, 6.3 x 1044 is written is standard form because 6.3 is between 1 and 10. 63 x 1044 is not in standard form because 63 does not lie between 1 and 10. Standard form is sometimes also called scientific notation. 8. Which of these numbers are in standard form? A. 4.5 x 107 B. 13 x 104 C. 0.9 x 10-2 D. 9.99 x 10-3 E. 4.5 billion F. 2.5 x 10 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 9. Write these numbers in standard form a. 87 000 b. 1 042 000 b. 1 394 000 000 d. 0.007 c. 0.000 002 84 f. 0.000 100 3 Q9 hint - Write the number between 1 and 10 first Then multiply by a power of 10. 10. Write these as ordinary numbers, a. 4 x 105 b. 3.5 x 102 7 7 -- Surds Surds Page Page 15 15 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 11. STEM / Reasoning a The distance from the Sun to Neptune is 4 500 000 000 000m. i. Write this number in standard form. ii. Enter the ordinary number in your calculator and press the = key. Compare your calculator number with the standard form number. Explain how your calculator displays a number in standard form. b. The thickness of a sheet of a paper is 0.000 07 m. i. Write this number in standard form. ii. Enter the ordinary number in your calculator and press the = key. Compare your calculator number with the standard form number. c. Reflect Why do you think that scientists use standard form for very large and very small numbers? 7 7 -- Surds Surds Page Page 15 15 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7

7 -- Surds Surds Page Page 15 15 Example 3 Work out (5 x 1033) x (7 x 1066) 5 x 7 x 1033 x 1066 35 x 1099 35 = 3.5 x 1011 Rewrite the multiplication grouping the numbers and the powers. Simplify using multiplication and the index law xm x xn= xmm+n This is not in standard form because 35 is not between 1 and 10. Write 35 in standard form. Work out the final answer. 10 34 x 1099 = 3.5 x 1011 x 1099 = 3.5 x 1010 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 12. Work out these. Use a calculator to check your answers. a. (3 x 102) x (2 x 105) b. (5 x 103) x (4 x 107) c. (8 x 10-2) x (6 x 107)

d. (8 x 106) (4 x 103) e. (9 x 10-2) (3 x 106) f. (2 x 103) (8 x 107) g. (5 x 103)2 h. (4 x 10-2)3 Q12g Q12g hint hint (5 (5 xx 10 1033))22 == (5 (5 xx 10 1033)) xx (5 (5 xx 10 1033)) 13. STEM / Problem-solving The Sun is a distance of 1.5 x 108 km from the Earth. Light travels at a speed of 3 x 10s km per second. How many seconds will it take for light from the Sun to reach the Earth? 7 7 -- Surds Surds Page Page 16 16 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

14. STEM / Problem-solving A water molecule has a mass of 3 x 10-29 kg. A bottle contains 1.7 x 1028 molecules of water. Calculate the mass of water in the bottle. 15. a. Write these numbers as ordinary numbers. i. 8 x 104 ii 3 x 102 b. Work out (8 x 104) + (3 x 102), giving your answer in standard form. Q15b Use Q15b hint hint Use your your answers answers from from part part aa to to write write the the answer answer as as an an ordinary ordinary number. number. Then Then convert convert this this to to standard standard form. form. 7 7 -- Surds Surds Page Page 16 16 1.6 Powers of 10 and Standard Form 1 1 -- Number Number 2 2 Place Place Value

Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 16. Work out these. Give your answers in standard form. a. 3.4 x 105 + 6.7 x 104 b. 9.8 x 104 - 2.2 x 102 c. 7.2 x 102 + 6.2 x 10-3 d. 8.3 x 105 - 7 x 10-1 17 Exam-Style Questions (7 x 10x) + (7 x 10y) + (7x10z) = 700 070.07 Write down a possible set of values for x, y and z. Exam hint (3 marks) Exam hint Dont Dont just just write write down down the the possible possible values values -give give your your working working to to show show how how you you worked worked out out the the values. values. 7 7 -- Surds

Surds Page Page 16 16 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 14 14 Objectives Why Learn This Understand the difference between rational and Surds are used to irrational numbers. express irrational Simplify a surd. numbers in exact form. Rationalise a denominator. Fluency What does the dot above the 1 mean in 0.1? What are the missing numbers? 147 = 3 x 125 = x 25

180=5 x 96 = x 16 ActiveLearn - Homework, practice and support: Higher 1.7 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 1. Warm Up Work out a. b. 2. Write each number as a fraction in its simplest form, a. 0.6 b. 0.85 c. 1.625 d. 4.25 e. 0.3 f. 1.5 3. Write to 2 decimal places a. b. b. d. Discussion: Which is more exact, the square root or the decimal? Key Point 13 A A surd

surd is is aa number number written written exactly exactly using using square square or or cube cube roots. roots. For For example example and and are are surds. surds. and and are are not not surds, surds, because because == 22 and and == 33 7 7 -- Surds Surds Page Page 16 16 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 4. a. Work out i. x ii. b. Work out i. ii. c. What do you notice about your answers to parts a and b? d. Find the missing numbers. i. x = ii. x = iii. = Key Point 14 = 7 7 -- Surds Surds Page Page 17 17 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Find the value of the integer k to simplify these surds. a. = k b. = k c. k

d. k Q5 Q5 communication communication hint hint -- An An integer integer is is aa positive positive or or negative negative whole whole number number or or zero zero 6. Simplify these surds a. b. c. d. e. 4 f. 6 Q6 Q6 hint hint Find Find aa factor factor that that is is also also aa square square number number 7 7 -- Surds Surds Page Page 17 17 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF &

& LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Use a calculator to work out a. As a simplified surd b. As a decimal Q7 hint Make sure you know how to switch between surd and decimals on your 8. a. calculator A surd simplifies to 4. What could the original surd be? b. Reflect How did you find Key Point 15 the surd? = 7 7 -- Surds Surds Page Page 17 17 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices

Indices 6 6 Powers Powers of of 10 10 9. Simplify a. = = b. c. d. = Key Point 15 . . 7 7 -- Surds Surds Page Page 17 17 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 10. Copy and complete the table using the numbers below. Rational Irrational

-4 1.4 0.3 11. Solve the equation x2 - 90 = 0, giving your answer as a surd in its simplest form. Discussion Can you solve the equation x2 + 90 = 0 in the same way? Explain your answer. Q11 hint x22 = x22 x2 = 7 7 -- Surds Surds Page Page 18 18 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 12. Solve these equations, giving your answer as a surd in its simplest form, a. 4x2 = 200 b. x2 = 80 c. 3x2 = 36 d. 2x2 -14 = 42 13. The area of a square is 60cm2. Find the length of one side of the square.

Give your answer as a surd in its simplest form. 14. a. Work out i. 5x4 ii. 4 x6 iii. 9 x 4 iv. 8 x3 Q14b.hint thetointegers together Use- Multiply a calculator check your to parts i to iv. and theanswers surds together. Simplify. 7 7 -- Surds Surds Page Page 18 18 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

7 7 -- Surds Surds Page Page 18 18 Key Point 17 To rationalise the denominator of , multiply by . Then the fraction will have an integer as the denominator. Example 4 Rationalise the denominator a. b. b. = x = = c. = = 5 = = x = = Multiplying by is the same as multiplying by 1, so this does not change the value. First simplify Simplify the fraction before rationalising. 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 15. Rationalise the denominators.

Simplify your answers if possible. a. b. c. d. e. f. g. h. 16. Reasoning / Problem-solving Ben types into his calculator. His display shows . a. Show that = b. Use your calculator to check your answers from Q14. 7 7 -- Surds Surds Page Page 18 18 1.7 - Surds 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7

7 -- Surds Surds Page Page 19 19 17. The area of a rectangle is 20 cm2. The length of one side is cm. Work out the length of the other side. Give your answer as a surd in its simplest form. 18. Work out the area of these shapes. Give your answer as a surd in its simplest form. Q18 hint - Multiply the numerators and denominators separately. Then rationalise the denominator. 1 Problem Solving 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Objectives Use pictures or lists to help you solve problems. Example 5 7 7 -- Surds Surds

Page Page 19 19 1 Problem Solving 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 1. There are 20 chairs in a conference room. The conference organiser can put 4,5 or 6 chairs at a table, a. Describe the different ways the room can be arranged so that all the chairs are used. b. What is the maximum number of tables required in the room? Q1 Q1 hint hint -- Draw Draw aa picture picture or or use use aa list. list. 2. A play park is 18 m wide and 31.5 m long. The council plans to enclose it with a fence, using a supporting post every 2.25 m. How many posts does the council need? Q2 Q2 hint

hint -- Draw Draw aa picture picture to to help help you you see see what what you you do do to to solve solve the the problem. problem. 7 7 -- Surds Surds Page Page 20 20 1 Problem Solving 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3. When two plant stakes are placed end to end, their total length is 1.45 m. When the two stakes are placed side by side, one is 0.15 m longer than the other. What lengths are the stakes?

Give your answer in cm. Q3 hint - Draw one picture to represent the first sentence. Draw another picture to represent the second sentence. 7 7 -- Surds Surds Page Page 20 20 1 Problem Solving 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 4. Finance In a canteen, a starter costs 0.80, a main costs 2.40 and a dessert costs 1.20. Three friends bought lunch and paid 10 in total. They each had at least two courses. How many starters, mains and desserts did they buy? Q4 hint - Find numbers that add to 10. 5. Finance A bicycle shop hires road bikes for 25 per day and tandems for 40 per day. One day a family pays 155. a Which type of bicycles did they hire? b How many people are in the family? 7

7 -- Surds Surds Page Page 20 20 1 Problem Solving 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. A tour company offers three different walking tours. The landmark tour leaves every 15 minutes. The parks tour leaves every 20 minutes. The museum tour leaves every 45 minutes. All walking tours start at 9am. When do the landmark, parks and museum tours next leave at the same time? Q6 Q6 hint hint -- List List all all the the different different times times each each tour tour leaves. leaves. 7. Reflect How can you solve Q6 without

making a list? Discussion Does it matter how you solve a maths problem? 7 7 -- Surds Surds Page Page 20 20 1 Checkup 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Check up Calculations, factors and multiples 1. 16.7 x 9.2 = 153.64 Use this fact to work out the calculations below. Check your answers using an approximate calculation, a. 167 x 9.2 b. 1.5364 1.67 2. Estimate the value of to the nearest tenth. 4. Write 90 as a product of its prime factors in index form. ActiveLearn - Homework, practice and support: Higher 1 Check up 7 7 -- Surds

Surds Page Page 20 20 1 Checkup 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3. a. Estimate i. ( - 6.17) x 1.98 ii. b. Use your calculator to work out each answer. Give your answers correct to 1 decimal place. 5. Find the highest common factor (HCF) and lowest common multiple (LCM) of 14 and 18. 6. In prime factor form, 2450 = 2 x 52 x 72 and 68 600 = 23 x 52 x 73. a. What is the HCF of 2450 and 68600? Give your answer in prime factor form, b. What is the LCM of 2450 and 68600? Give your answer in prime factor form. 7 7 -- Surds Surds Page Page 20 20

1 Checkup 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Indices and surds 7. Copy and complete. a. 10 =1000 c. 2 =16 d. 5 = 1 8. Work out a. 25) x 3]2 b. 43 = b. [(62 = 9. Write each product as a single power. a. 9-3 x 97 b. 27 x 9 x 27 c. 57 52 d. d. (24)3 f. (42)-1 7 7 -- Surds Surds Page Page 21 21 1 Checkup 1 1 -- Number Number 2 2

Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Indices and surds 10. Work out a. 2-4 253/2 c. ()3/4 b. d. 16- 11. Simplify a. b. 5 12. Rationalise the denominators. Simplify your answers if possible. 7 7 -- Surds Surds Page Page 21 21 1 Checkup 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF

HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Standard form 13. Write these numbers in standard form, a. 32 040 000 b. 0.0007 14. Write these as ordinary numbers. a. 5.6 x 104 b. 1.09 x 10-3 15. Work out these. Give your answers in standard form. a. (5 x 104) x (9 x 107) b. (3 x 108) (6 x 105) 7 7 -- Surds Surds Page Page 21 21 1 Checkup 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices

Indices 6 6 Powers Powers of of 10 10 Feeling doubtful Confident What next? Use your results to decide whether to strengthen or extend your learning. Reflect Standard form How sure are you of your answers? Were you mostly Just guessing 7 7 -- Surds Surds Page Page 21 21 1 Challenge 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of

of 10 10 17. The diagram shows a warehouse (W) and five destinations (A, B, C, D, E), and the times it takes to drive between each of them. 18. A delivery driver has to deliver packages to A, B, C, D and E. He starts and ends at W. a. Which is the quickest route? b. Write your own delivery driver question. 7 7 -- Surds Surds Page Page 21 21 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Calculations, factors and multiples 1. Copy and complete these number patterns. a. 0.38 x 29.4 = 11.172 3.8 x 29.4 = 38 x 29.4 = 380 x 29.4 = 3800 x 29.4= b. 6011.545 94.67 = 63.5

60115.45 94.67 = 601154.5 * 94.67 = 6011545 94.67 = 60115 450 94.67 = Q1a hint - 3.8 = 0.38 x 10 3.8 x 29.4 = 11.172 x 10 7 7 -- Surds Surds Page Page 22 22 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 2. 8.9 x 7.21 = 64.169 Use this fact to work out the calculations below. Check your answers using an approximate calculation, a. 8.9 x 72.1 b. 8.9 x 7210 c. 0.89 x 0.721 d. 0.089 x 0.721 Q2 hint - Write out a number pattern to e. 64.169 72.1 f. 6416.9 help you. 7.21

3. Copy and complete this square root number line. 7 7 -- Surds Surds Page Page 22 22 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 4. Estimate the value to the nearest tenth. a. b. c. Q4a Q4a hint hint -- Use Use your your number number line line from from Q3. Q3. Which Which two two square square roots roots does

does lie lie between? between? Which Which is is itit closer closer to? to? 5. a. Estimate 1. 25.76 2. + Q5a Q5a hint hint -- Round Round each each number number to to the the nearest nearest whole whole number. number. Which Which square square root root is is itit closest closest to to on on your your square square root root number number line? line? 7 7 -- Surds Surds Page Page 22 22 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value

Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. Copy and complete these calculations in index form a. 2 x 2 x 2 x 3 x 3 = 23 x 3 b. 2 x 2 x 3 x 5 = 2 x x c. 3 x 3 x 3 x 3 x 7 x 7 = 7. a. Copy and complete this factor tree for 60 until you end up with just prime factors. b. Write 60 as a product of its prime factors. c. Write your answer to part b in index form. Q5a Q5a hint hint -- Write Write all all the the prime prime factors factors from from your your tree tree multiplied multiplied together.7b together.7b 7 7 -- Surds Surds Page Page 22 22 1 Strengthen

1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 8. Write each number as a product of its prime factors in index form, a. 24 b. 80 c. 45 d. 30 e. 16 f. 72 10.Find the highest common factor (HCF) and lowest common a. 20 and 80 b. 21 and Q10 hint 28 - Use the method in Q9. c. 15 and 25 d. 44 and ActiveLearn - Homework, practice 36 and support: Higher 1 - Strengths and support: Higher 1 - Strengths 7 7 -- Surds

Surds Page Page 22 22 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 9. a. Write 18 as a product of its prime factors. 18 = x x b. Write 45 as a product of its prime factors, c. Copy and complete this diagram. i. Put any prime factors of both numbers where circles overlap. ii. Put the remaining prime factors of 18 in the left-hand part of the left circle. iii. Put the remaining prime factors of 45 in the right-hand part of the right circle. d. Work out the HCF. e. Work out the LCM. 7 7 -- Surds Surds Page Page 23 23

1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Indices and Surds 1. Copy and complete. a. 2 x 2 x 2 = 2 = b. 5 x 5 = 5 = c. -3 x -3 x -3 = (-3) = 2. Work out a. 25 = b. 53 = c. 4 =64 d. 10 = 1000 Q2c hint - 411 = 4 422 = 4 x 4 = 433 = 4 x 4 x4 = 7 7 -- Surds Surds Page Page 23 23 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3

3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3. Work out a. 3 x b. 23 x c. x d. 102 e. (-3)2 x f. x x Q3 hint - First work out any powers or roots. Then multiply or divide. 4. Work out a. 5 x (42 - 32) - 23 b. [(92 32) + 22]2 Q4b Q4b hint hint -- First First work work out out the the round round brackets. brackets. Then Then work work out out the the square square brackets. brackets. 7 7 -- Surds Surds Page Page 23 23 1 Strengthen

1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Copy and complete a. (3 x 3 x 3 x 3 x 3) x (3 x 3 x 3) = 3 x 3 = 3 b. (4 x 4 x 4 x 4) x (4 x 4 x 4 x 4 x 4 x 4) = 4 x4 = 4 c. = = 6 d. = = 7 e. To multiply powers, ______ the indices. To divide powers, ______ the indices. 7 7 -- Surds Surds Page Page 23 23 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3

HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 6. Work out a. 56 x 53 = b. 72 x 79 = c. 58 53 = d. 98 92 = e. 84 x 8-6 = f. 73 7-4 = 7. Write these as a single power of a prime number, a. 16 x 8 = 2 x 2 = b. 25 x 125 x 25 c. 16 x 64 x 8 Q6 hint - Use the rules from Q5. d. 27 x 27 x 9 7 7 -- Surds Surds Page Page 24 24 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF

HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 8. Copy and complete, a. (42)3 = 42 x 42 x 42 = 4 b. (63)4 = 63 x 63 x 63 x 63 = 6 c. (75)2 = d. (83)7 = e. To work out a power raised to a power, __________ the indices. 9. a. Work out using a calculator. i. 5 ii 7 iii 192 iv (-3) b. Use your answer to part a to work out these without a 7 7 -- Surds Surds Page Page 24 24 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5

5 Indices Indices 6 6 Powers Powers of of 10 10 10. a. b. c. d. e. Work out using a calculator. i. ii 169 Use your answer to part a to work these out without a calculator, i. 64 ii 25 iii 81 iv 144 Work out using a calculator, i. ii 5121/3 Use your answer to part c to work these out without a calculator, i. 1251/3 ii 271/3 ii. 10001/3 iv 81/3 Copy and complete. 161/4 = = Q10e hint - Use what you have learned in parts a to d. 7 7 -- Surds Surds Page Page 24 24 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF

HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 11. a. Work out i. 64 ii 642/3 2/3 = (64 )22 Work out Q11a i hint 642/3 1/3 and square your answer. 641/3 b. Use your answer to Q10d to help you work out i. 125? ii 27? iii. 1000? iv 8? c. Use your answer to Q10e to 3/4 help you work out 16 . Q11a i hint 16 =(16) Use the same strategy as in part a. 7 7 -- Surds Surds Page Page 24 24 1 Strengthen 1 1 -- Number

Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 24 24 12. a. Copy and complete. i. 4-3 = ii. = 10 iii. = 2 iv. 3- = v. () = b. Work out i. 4-1 ii. 10-2 iii. 36 iv. 125- 13. Copy and complete. a. 50 = x 2, so = x = b. 84 = x 21, so = x = Simplify c. d. e. Q13c Q13c hint hint -- Write Write the the square square numbers numbers up up to to 100. 100. Find Find aa square

square number number that that is is aa factor factor of of 96. 96. 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 14. Work out a. x b. x c. x 15. Rationalise the denominator. Simplify your answer if possible. a. b. c. d. Q15a hint multiply both the numerator and denominator by Q11a i hint First rewrite the denominator. = 7 7 -- Surds Surds Page Page 26 26 1 Strengthen

1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 Standard form 1. Are these numbers in standard form? If not, give reasons why. a. 9.004 x 10-3 b. 32 x 105 c. 7.3 million d. 0.8 x 107 2. Write each number using standard form, a. 68 000 = 6.8 x 10 b. 94 000 000 c. 801 000 d. 0.000 004 e. 0.0039 7 7 -- Surds Surds Page Page 25 25 Q2a hint 6.8 6.8 lies lies between 1 and 10. Multiply Multiply by by 10

10 how how many many times to to get get 68 68 000? 000? Q2d Q2d hint hint 44 lies lies between between 1 and and 10. 10. Divide Divide by by 10 10 how how many many times times to to get get 0.000 0.000 004? 004? 1 Strengthen 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3.

Work out a. (4 x102) x (2 x 107) = x 10 b. (3 x 109) x (2 x 105) c. (6 x 104) x (1 x 10-2) d. (6 x 105) x (8 x 104) e. (7 x 103) x (8 x 106) f. (8 x 10-4) x (6 x 10-2) 4. a. Write 2.5 x 104 and 1.3 x 10-2 as ordinary numbers, b. Use your answers to part a to help you work out (2.5 x 104) + (1.3 x 10-2) Q3d Q3d hint hint 48 48 == 4.8 4.8 xx 10 10 Q4a Q4a hint hint 10 1044 == 10000 10000 10 10-2-2 == 7 7 -- Surds Surds Page Page 25 25 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers

Powers of of 10 10 1. Problem-solving Square A has a side length of 9.2 cm. Square B has a perimeter of 34.4 cm. Square C has an area of 80 cm2, a. Which square has the greatest perimeter? b. Which square has the smallest area? 2. Show that 272 = 93 = 36. Q2 communication hint - Show that' means show your working. 7 7 -- Surds Surds Page Page 25 25 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 3 Exam-Style Questions Here are some properties of a number. It is a common factor of 216 and 540.

It is a common multiple of 9 and 12. Write two numbers with these properties. Exam hint (6 marks There are 6 marks so most of them are for showing your working 7 7 -- Surds Surds Page Page 25 25 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 4. a. Write 48,90 and 150 as products of their prime factors, b. Use a Venn diagram to work out the HCF and LCM of 48,90 and 150. Discussion Explain how the diagram can be used to find the HCF and LCM of any two of

the numbers. 7 7 -- Surds Surds Page Page 26 26 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 5. Real A new school is deciding whether their lessons should be 30,50 or 60 minutes. Each length of lesson fits exactly into the total teaching time of the school day. How long is the teaching time of the school day? Discussion Ryan says there is more than one answer to this question. Is Ryan correct? Explain your answer. 6. Reasoning a. Use prime factors to explain why numbers ending In a zero must be divisible by Z and 5. b. How many zeros are there at the end of 24 x 37 x 56 x 72? c. Use prime factors to work out 32 x 9 x 3125. Write your answer as an ordinary number and in standard form. 7 7 -- Surds

Surds Page Page 26 26 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Write each of these as a simplified product of powers. a. 105 x 23 x 5 = (2 x 5)5 x 23 x 5 = 2 x 5 x 23 x 5 = 2 x 5 b. 63 x 24 x 33 c. 153 x 104 x 62 d. 30x24*xl53 8. Estimate the value of 5.14 Q8 hint - 5.144 = 5.122 x 5.122 7 7 -- Surds Surds Page Page 26 26 1 Extend 1 1 -- Number Number 2

2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 9. STEM Write each answer i. as an ordinary number ii. in standard form, a. Saturn has a diameter of 120 536 000m. Convert this to kilometres. b. The distance from the Sun to Mars is 227 900 000km. Convert this distance to metres. c. The diameter of a grain of sand is 4 pm. Convert this to metres, d. The wavelength of an X-ray is 0.1 nm. Convert this to metres. 7 7 -- Surds Surds Page Page 26 26 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM

LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 10. Every six months, new licence plates are issued in the UK. A licence plate consists of two letters, then two numbers, then three letters. The numbers are fixed, but the letters vary. a. If all letters can be used, how many possible combinations are there? b. If only 21 letters can be used, how may possible combinations are there? 7 7 -- Surds Surds Page Page 26 26 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 11. STEM / Problem-solving A container ship carries 1.8 108kg. An aeroplane can carry 3.8 x 105kg. What is the difference in their mass? Write your answer in standard form. 12 Exam-Style Questions Work out a. + b. Write each answer in the form a. (3 marks) Q12 Q12 strategy strategy hint hint -- To To add add and and subtract subtract fractions fractions you you need need to to write write them them with with aa common common denominator. denominator. 7 7 -- Surds Surds Page Page 27 27 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers

Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 13. Write 3- as a surd and rationalise the denominator. 14 Exam-Style Questions A restaurant offers 5 starters, 7 mains and 3 desserts. A customer can choose just one course any combination of two courses all three courses. Show that a customer has 191 options altogether. Exam hint (3 marks) Show your working clearly 7 7 -- Surds Surds Page Page 27 27 1 Extend 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6

Powers Powers of of 10 10 15. Estimate the value of a. (3.1 X 103)2 b. (1.9 x 10-2)3 Q15a hint (3.1 x 1033)22 = 3.1 x 1033 x 3.1 x 1033 16. Estimate to the nearest whole number a. 2.73 b. 1.43 c. 2.14 d. e. f. 7 7 -- Surds Surds Page Page 27 27 1 Knowledge Check 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10

7 7 -- Surds Surds Page Page 27 27 When there are m ways of doing one task and n ways of doing a different task, the total number of ways the two tasks can be done Mastery lesson 1.1 is m x n. You can round numbers to 1 or 2 significant figures to estimate the answers to calculations, including calculations with powers and Mastery lesson 1.2 roots You can use a prime factor tree to write a number as the product Mastery lesson 1.3 of its prime factors You can use a Venn diagram of prime factors to work out the highest common factor and lowest common multiple of two Mastery lesson 1.3 numbers The prime factor decomposition of a number is the no. written as the product of its prime factors. It is usually written in index form When multiplying powers, add the indices: xm x xn = xm+n Mastery lesson 1.3 When dividing powers, subtract the indices: xm xn = xm-n Mastery lesson To raise a power to another power, multiply the indices. 1.4, 1.5 x-n = x1/n = xm/n = ()m A number in standard form is written in the formats A x 10n, where A is a number between 1 and 10 and n is an integer. Mastery lesson 1.6 1 Knowledge Check 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers

Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 28 28 To write a number in standard form: o work out the value of A o work out how many times A must be multiplied or divided Mastery lesson 1.6 by 10. Mastery lesson 1.7 This is the value of n. To simplify a surd, identify any factors that are square numbers To rationalise a denominator, multiply the numerator and the denominator by the surd in the denominator and simplify. For each statement A, B and C, choose a score: Mastery lesson 1.8 1 - strongly disagree; 2 - d isagree; 3 - agree; 4 - strongly agree A. I always try hard in mathematics B. Doing mathematics never makes me worried C. l am good at mathematics For any statement you scored less than 3, write down two things you could do so that you agree more strongly i n the future. 1 Unit Test 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6

Powers Powers of of 10 10 1. 6.23 x 5.4 = 33.642 a. Write down two more multiplications with an answer of 33.642. b. Write down a division with an answer of 0.623. (3 marks) 2 Exam-Style Questions List these numbers in order, starting with the smallest. Show your working. 3.22 13.74 (3 marks) 3. a. Estimate (17.9 - ) x 3.89 b. Use a calculator to work out the answer. Give your answer correct to 1 decimal place. (2 marks) 7 7 -- Surds Surds Page Page 28 28 1 Unit Test 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10

10 4. a. Write 42 as a product of its prime factors. b. Use your answer to write 843 as a product of its prime factors in index form. (4 marks) 5. Workout the HCF and LCM of 75 and 30. (3 marks) 6. Real Ben and Sadie are doing a sponsored walk around a circuit. Ben takes 25 minutes to do one circuit and Sadie takes 45 minutes. They start together at 9:30 am. When will they next cross the start 7 7 -- Surds Surds Page Page 28 28 1 Unit Test 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7. Find the value of a. a. 53 x 5a = 59 b. b. 6a + 6-5 = 68

c. 8a x 8a = 84 (3 marks) 8. Write (38)4 as a single power. (1 mark) 9. Use prime factors to determine whether 2520 is divisible by 18. (2 marks) 7 7 -- Surds Surds Page Page 28 28 1 Unit Test 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 11. Write ()-2 as a fraction in its simplest form. (2 marks) 12. Let x = 6 x 105 and y = 8 x 104. Work out a. x + y b. x - y Write your answers in standard form. 13. 918 = 27x Work out the value of x. (2 marks) 14. Work out the area of this shape. Write your answer as a

7 7 -- Surds Surds Page Page 28 28 1 Unit Test 1 1 -- Number Number 2 2 Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 15 Exam-Style Questions How many different 4-digit odd numbers are there, where the first digit is not zero? (3 marks) 16. Rationalise the denominator. 7 7 -- Surds Surds Page Page 29 29 1 Unit Test 1 1 -- Number Number 2 2

Place Place Value Value 3 3 HCF HCF & & LCM LCM 4 4 -- Powers Powers 5 5 Indices Indices 6 6 Powers Powers of of 10 10 7 7 -- Surds Surds Page Page 28 28 Sample student answer The maths is correct, but the student will only get 2 marks. Why? 15 Exam-Style Questions One sheet of paper is 9 x 10-!cm thick. Mark wants to put 500 sheets of paper into the paper tray of his printer. The paper tray is 4 cm deep. Is the paper tray deep enough for 500 sheets of paper? You must explain your answer. (3 marks) June 2013, Q15, 1MA0 / 1H Student answer 9 x 10-3 x 500 0.009 x 500 9 x 500 = 4500 + 1000 = 4.5

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