MZQ8_ch01body.fm Page 17 Wednesday, July 23, 2003 6:33 PM Prime numbers We give special names to numbers depending on how many factors they have. A prime number has exactly two factors: itself and 1. A composite number has more than two factors. 1 is a special number—neither prime nor composite. Prime factors and factor trees A prime factor of a number is a factor that is also prime. Every composite number can be expressed as a product of prime numbers. A factor tree is a good way to find these prime factors. worked example 10 Write 24 as a product of prime factors. Steps Solution 1. Break the number into a pair of factors 24 24 (not using 1), then break each subsequent factor into a pair of factors. 12 2 83 Circle any prime factors as they or appear, and stop the branch at this 6 2 4 2 point. (Two possible trees are shown.) 32 22 2. Write the number as a product of the 24 = 2 × 2 × 2 × 3 circled factors. e eTutorial Powers e eTutorial Powers are a short way of writing repeated factors. 52 is an example of a power. The small 2 is the index and tells us index how many factors of 5 there are. The 5 is called the base. base 2 For example: 42 = 4 × 4 5 53 = 5 × 5 × 5 power 74 = 7 × 7 × 7 × 7 95 = 9 × 9 × 9 × 9 × 9 1 ● whole NUMBERS 17 MZQ8_ch01body.fm Page 18 Wednesday, July 23, 2003 6:33 PM For 42 we say ‘four squared’ or ‘four to the power of two’ or ‘four to the Power Man to base … second power’. can you hear me? For 53 we say ‘five cubed’ or ‘five to the power of three’ or ‘five to the third power’. For 74 we say ‘seven to the power of four’ or just ‘seven to the fourth’. Another word for index is exponent. Numbers written using index notation are in index form or exponential form. 43 is in index form. 4 × 4 × 4 is in expanded form. worked example 11 (a) Write out 43 in expanded form and work out the answer. (b) Write out 74 in expanded form and work out the answer. Steps Solutions (a) 1. Write 43 in expanded form. (a) 43 = 4 × 4 × 4 2. Write down the answer. = 64 (b) 1. Write 74 in expanded form. (b) 74 = 7 × 7 × 7 × 7 2. Write down the answer. = 2401 e eTutorial Powers on calculators Many calculators have a key to help you work quickly with powers, usually xy or ^ . For example, if you want to work out 123, press 1 2 xy 3 = . To work out 52, simply press 5 x 2 . Squares and square roots A number is called a perfect square if it can be found by multiplying another number by itself (i.e. by squaring the other number). Finding the square root is the opposite to squaring a number. For example, to find the square root of 16, you have to think of a number that when multiplied by itself gives you 16. The number is 4 since 4 × 4 = 16. This means the square root of 16 is 4. We write this as 16 = 4. You can use your calculator to help with some calculations. Your calculator will have a key like this: . eTutorial To find 2304 , press 2 3 0 4 = e e eTutorial or 2 3 0 4 depending on the type of calculator. Then read off the answer: 48. e eTutorial 18 Heinemann MATHS ZONE MZQ8_ch01body.fm Page 19 Wednesday, July 23, 2003 6:33 PM exercise 1.5 Primes, powers and square roots Preparation: Exercise 1.4 1 Complete these factor trees, then write each number as a product of its prime factors. (a) 16 (b) 50 e Worksheet C1.8 82 16 = × × × 50 = × × (c) 100 (d) 96 10 3 100 = × × × 96 = × × × × × 2 Express each number as a product of its prime factors. e Interactive (a) 8 (b) 12 (c) 18 (d) 26 e Hint (e) 20 (f) 48 (g) 68 (h) 44 e eQuestions (i) 64 (j) 72 (k) 108 (l) 144 (m) 140 (n) 1000 (o) 800 (p) 400 e eQuestions 3 Find three different numbers that can each be expressed as a product of e Worksheet C1.9 four prime factors. (The product may contain repeated factors.) 4 Simplify these with your calculator, using the shortest method possible. (a) 212 (b) 182 (c) 482 (d) 612 e Hint (e) 752 (f) 882 (g) 5402 (h) 4392 (i) 23052 (j) 17662 (k) 432 − 203 (l) 742 − 1341 (m) 682 × 142 (n) 372 × 222 (o) 262 + 412 (p) 232 − 142 5 What number do you have to square to get each of these? (a) 9 (b) 4 (c) 49 (d) 64 e Hint (e) 81 (f) 25 (g) 144 (h) 10 000 e eQuestions 6 Evaluate the following. (a)25 (b) 49 What number times itself gives 25? (c)36 (d) 64 (e)100 (f) 121 (g)196 (h) 169 (i)1 (j) 0 (k)4900 (l) 400 (m)1600 (n) 2500 (o)360 000 (p) 1 000 000 e eQuestions 7 Write each of these in index form. (Don’t work out the answer.) (a) 8 × 8 × 8 (b) 4 × 4 × 4 × 4 × 4 × 4 e Hint (c) 9 × 9 × 9 × 9 (d) 7 × 7 × 7 1 ● whole NUMBERS 19 MZQ8_ch01body.fm Page 20 Wednesday, July 23, 2003 6:33 PM (e) 5 × 5 × 5 × 5 × 5 × 5 (f) 9 × 9 × 9 × 9 × 9 (g) 12 × 12 × 12 × 12 × 12 (h) 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 (i) 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 (j) 11 × 11 × 11 × 11 × 11 × 11 × 11 × 11 (k) seventeen cubed (l) thirteen to the power of seven (m) eight to the fourth power (n) nine to the sixth (o) eleven to the seventh (p) nine to the third e eQuestions 8 Write each of these out in expanded form, then work out the answer. (a) 23 (b) 24 (c) 26 (d) 25 e Hint (e) 18 (f) 07 (g) 06 (h) 17 (i) 103 (j) 105 (k) 64 (l) 84 (m) 113 (n) 124 (o) 46 (p) 66 9 What number do you have to cube to get each of these? (a) 1 (b) 27 (c) 8 (d) 64 e Hint (e) 8000 (f) 1000 (g) 125 (h) 1 000 000 e eQuestions 10 3 means cube root. The cube root of 27 is 3 because 3 × 3 × 3 = 27. Evaluate, without using a calculator. (a)3 8 (b)3 64 (c)3 1000 (d) 3 1 e Hint (e)3 0 (f)3 125 (g)3 8000 (h) 3 27 000 11 (a) Arrange these numbers in ascending order (smallest to largest). e Hint 45, 54, 1200, 103, 46, 55 (b) Arrange these numbers in descending order (largest to smallest). 1002, 105, 11000, 0100, 32, 23 12 (a) Copy and complete: 101 = 10 102 = 10 × 10 = 100 103 = 10 × 10 × 10 = 1000 104 = 105 = (b) The number 10100 was called a googol by the mathematician Edward Kasner. (i) How many times is 10 multiplied by itself to get a googol? (ii) Look at the pattern in part (a). If you were to work out what a googol is equal to, how many zeros would follow the 1? (c) If you raise the number 10 to the power of a googol, you get a e hi.com.au number called a googolplex. (i) How many times is 10 multiplied by itself to get a googolplex? (ii) If you were to work out what a googolplex is equal to, how many zeros would follow the 1? How much time do you think you save by writing a googolplex in index form? 20 Heinemann MATHS ZONE MZQ8_ch01body.fm Page 21 Wednesday, July 23, 2003 6:33 PM 13 Without using your calculator, find between which two consecutive whole numbers each of the following lies. (a)10 (b)5 (c)20 (d) 62 e Hint (e)99 (f)2 (g)70 (h) 108 14 Without using your calculator, find between which two consecutive whole numbers each of the following lies. (a)3 10 (b)3 52 (c)3 1001 (d) 3 30 e Hint (e)3 5 (f)3 71 (g)3 120 (h) 3 2 15 (a) Use your calculator to answer TRUE or FALSE to each statement. (i) 46 is bigger than 64. (ii) 210 is bigger than 102. e Hint (iii) 39 is bigger than 93. (iv) 192 is bigger than 219. (b) Look at your answers for part (a) and without using your calculator answer TRUE or FALSE to each of these statements. (i) 98 is bigger than 89. (ii) 2100 is bigger than 1002. 16 Use your calculator to find the following through trial and error. (a) A number that when multiplied by itself three times gives 343 e Hint (b) A number that when multiplied by itself three times gives 1728 (c) A number that when multiplied by itself three times gives 2744 (d) A number that when multiplied by itself three times gives 39 304 e Worksheet C1.10 (e) A number that when multiplied by itself four times gives 4096 (f) A number that when multiplied by itself five times gives 161 051 17 (a) Copy the following pattern and complete it using your calculator. e Hint 112 = 1112 = 11112 = (b) Look at the pattern in part (a) and without using a calculator copy and complete the following.