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25 INDICES, STANDARD FORM AND SURDS O T LL I T D S E I S L E P R - UP I H S BE The photo shows a male Escheria coli bacteria. You may have heard of e-coli. These bacteria are commonly known in relation to food poisoning as they can cause serious illness. Each bacterium is about a millionth of a metre long. That can be written as 0.000001m, or in standard form as 1 × 106m long. Standard form allows us to write both very large and very small numbers in a more useful form. 610 Objectives Before you start In this chapter you will: work out the value of an expression with zero, negative or fractional indices convert between standard form and ordinary numbers calculate with numbers in standard form manipulate surds make estimates to calculations using numbers in standard form. You need to be able to: use the index laws round numbers to one significant figure. 25.1 Using zero and negative powers 25.1 Using zero and negative powers Objectives Why do this? You know that n0 1 when n ≠ 0. You know the meaning of negative indices. If you are x metres from a live band, the volume of sound they are producing is directly proportional to x2. This means that if you halve your distance to the band, the music will get four times as loud. Get Ready Work out 1. 32 2. 25 3. 43 Key Points For non-zero values of a a0 1 For any number n 1 an __ an Example 1 Work out the value of a 30 b 5–1 c 6–2 –2 d __2 5 ( ) a 30 1 Any number to the power of zero is 1. 1 b 51 __ 5 1 Use the rule an __ an 1 c 62 __ 6 2 1 __ 36 d (5 ) 2 __ 2 62 6 6 36. Examiner’s Tip 1 2 2 __ 5 5 2 __ 2 25 ___ 4 ( ) ( ) To work out the reciprocal of a fraction, turn the fraction upside down. Square the number on the top and the number on the bottom of the fraction. Do not change the fraction to a decimal. It is much easier to square the numbers in a fraction than it is to square a decimal. 611 Chapter 25 Indices, standard form and surds Exercise 25A B 1 2 Questions in this chapter are targeted at the grades indicated. Write down the value of these expressions. a 70 b 81 3 e (2) f 92 i (3)2 j (8)0 c 51 g 104 k 160 Work out the value of these expressions. a (_13 )1 b (_27 )1 c e (0.25)2 f g (1 ) j i _2 2 5 (_25 )3 (1 _13 )3 (_17 )2 (_53 )0 k (0.1)4 d 40 h 1450 l 106 d h (_14 )3 (_95 )1 l (0.2)3 25.2 Using standard form Objectives Why do this? You can convert ordinary numbers into standard form. You can convert numbers in standard form into ordinary numbers. You can calculate with numbers in standard form You can convert to standard form to make sensible estimates for calculations. Astronomers use standard form to record large measurements. The Sun’s diameter is about 1.392 106 km. Biologists working with microorganisms sometimes use standard form to record their very small sizes, like 2.1 104 cm. Get Ready 1. Work out a 103 b 10–2 2. Write 10 000 as a power of 10. 3. Work out 2.35 10 000. Key Points Standard form is used to represent very large (or very small) numbers. A number is in standard form when it is in the form a 10n where 1 a 10 and n is an integer. To use standard form you need to know how to write powers of ten in index form. 10 101 100 10 10 102 1000 10 10 10 103 A number in standard form looks like this. 6.7 104 This part is written as a This part is written as a number between 1 and 10. power of 10. 2 8 These numbers are all in standard form – 4.5 10 , 9 10 , 1.2657 106. These numbers are not in standard form – 67 109, 0.087 103 – because the first number is not between 1 and 10. It is often easier to multiply and divide very large or very small numbers, or estimate a calculation if the numbers are written in standard form. To input numbers in standard form into your calculator, use the 10 or EXP key. To enter 4.5 107 press the keys 4 · 5 � 10 7 . 612 standard form integer 25.2 Using standard form Example 2 Write these numbers in standard form. a 50 000 b 34 600 000 c 682.5 a 50 000 5 10 000 5 104 b 34 600 000 3.46 10 000 000 3.46 107 c 682.5 6.825 100 6.825 102 Use 3.46 not 34.6 or 346 as 3.46 is between 1 and 10. Write as an ordinary number a 8.1 105 Example 3 b 6 108 a 8.1 105 8.1 100 000 810 000 b 6 108 6 100 000 000 600 000 000 Exercise 25B 1 2 3 4 B Write these numbers in standard form. a 700 000 b 600 c 2000 d 900 000 000 e 80 000 Write these as ordinary numbers. a 6 105 b 1 104 c 8 105 d 3 108 e 7 101 Write these numbers in standard form. a 43 000 b 561 000 c 56 d 34.7 e 60 Write these as ordinary numbers. a 3.96 104 b 6.8 107 c 8.02 103 d 5.7 101 e 9.23 100 5 In 2008 there were approximately 7 000 000 000 people in the world. Write this number in standard form. 6 The circumference of Earth is approximately 40 000 km. Write this number in standard form. Example 4 Write in standard form a 0.000 000 006 b 0.000 56 a 0.000 000 006 6 0.000 000 001 1 6 __________ 1 000 000 000 9 1 6 __ 10 1 0.000 000 001 is equivalent to ___________ . 1 000 000 000 1 Using an __ an 6 109 b 0.000 56 5.6 0.0001 1 5.6 _____ 10 000 4 1 5.6 __ 10 Use 5.6 rather than 56 as 5.6 is between 1 and 10. 5.6 104 613 Chapter 25 Indices, standard form and surds Example 5 Write as an ordinary number a 3 106 b 1.5 103 3 a 3 106 ___ 6 1.5 b 1.5 103 ___ 3 10 10 3 _______ 1 000 000 15 ____ 1000 0.000 003 0.0015 Exercise 25C B 1 2 3 4 5 6 Write these numbers in standard form. a 0.005 b 0.04 c 0.000 007 d 0.9 e 0.0008 Write these as ordinary numbers. a 6 105 b 8 102 c 5 107 d 3 101 e 1 108 Write these numbers in standard form. a 0.0047 b 0.987 c 0.000 803 4 d 0.000 15 e 0.601 Write these as ordinary numbers. a 8.43 105 b 2.01 102 c 4.2 107 d 7.854 101 e 9.4 104 Write these numbers in standard form. a 457 000 b 0.0023 f 89 000 g 200 c 0.0003 h 0.005 26 d 2 356 000 i 6034 e 0.782 j 0.000 008 73 Write these as ordinary numbers. a 4.12 104 b 3 103 1 f 7.5 10 g 1.5623 102 c 2.065 107 h 5.12 107 d 4 106 i 2.7 105 e 3.27 108 j 6.12 101 7 1 micron is 0.000 001 of a metre. Write down the size of a micron, in metres, in standard form. 8 A particle of sand has a diameter of 0.0625 mm. Write this number in standard form. Example 6 Write in standard form a 40 102 b 0.008 102 Method 1 a 40 102 4 101 102 12 4 10 4 103 b 0.008 10–2 8 103 102 8 103 2 8 105 614 Examiner’s Tip Write 40 in standard form. Use the rule am an amn. Write 0.008 in standard form. Use am an amn. The power of 10 tells you how many 0s there are. 102 100 2 zeros 102 0.01 2 zeros 25.2 Using standard form Method 2 a 40 102 40 100 4000 4 103 1 b 0.008 102 0.008 ___ 100 Work out the calculation. Change the answer into standard form. 0.008 100 0.000 08 8 105 1 Use the rule an __ . an 1 ___ Multiplying by 100 is the same as dividing by 100. Exercise 25D 1 2 B Write these in standard form. b 980 103 a 45 103 c 3400 102 d 186 1010 Write these in standard form. a 0.009 105 b 0.045 106 c 0.3708 1012 d 0.006 107 3 Some of these numbers are not in standard form. If a number is in standard form then say so. If a number is not in standard form then rewrite it so that it is in standard form. a 7.8 104 b 890 106 c 13.2 105 d 0.56 109 7 10 8 e 60 000 10 f 8.901 10 g 0.040 05 10 h 9080 1015 i 6.002 105 j 0.0046 108 k 67 000 103 l 0.004 103 4 Write these numbers in order of size. Start with the smallest number. 6.3 106, 0.637 107, 6290000, 63.4 105 5 Write these numbers in order of size. Start with the smallest number. 0.034 102, 3.35 105, 0.000033, 37 104 Example 7 Work out (3 106) (4 103) giving your answer in standard form. (3 106) (4 103) 3 4 106 103 12 109 1.2 101 109 1.2 1010 Example 8 Rearrange the expression so the powers of 10 are together. Multiply the numbers. Use am an amn to multiply the powers of 10. 12 109 is not in standard form. Write your final answer in standard form. By writing 760 000 000 and 0.000 19 in standard form correct to one significant figure, work out an approximation for 760 000 000 0.000 19. 760 000 000 8 108 correct to one significant figure. 0.000 19 2 104 correct to one significant figure. 8 760 000 000 _________ 8 10 _____________ 0.000 19 2 10 4 8 10 8 _____ __ 2 10 4 1084 4 4 1012 Rearrange the expression so the powers of 10 are together. Divide the numbers. Use am an amn to divide the powers of 10. 615 Chapter 25 Indices, standard form and surds Exercise 25E A 1 2 3 AO3 AO2 AO3 Work out and give your answer in standard form. a (4 108) (2 103) b (6 105) (1.5 103) 7 6 d (6 10 ) (3 10 ) e (6 109) (5 103) c (4 107) (3 105) f (5 108) (2 103) Work out and give your answer in standard form. a (4 108) (2 103) b (9 105) (2 104) 8 13 d (8.6 10 ) (2 10 ) e (1 1012) (4 103) c (3 109) (6 103) f (7 109) (7 105) Express in standard form. a (2 105)2 b (5 105)2 c (4 106)2 d (7 108)2 4 By writing these numbers in standard form correct to one significant figure, work out an estimate of the value of these expressions. Give your answer in standard form. a 600 008 598 b 78 018 4180 c 699 008 198 d 8 104 660 000 0.000 078 5 Light travels at 3 108 metres per second. Work out the time it takes light to travel: a 200 metres b 1.5 centimetres. 6 The base of a microchip is in the shape of a rectangle. Its length is 2 103 mm and its width is 1.6 103 mm. Find the area of the base. Give your answer in mm2 in standard form. 7 The distance of the Earth from the Sun is approximately 93 000 000 miles. Light travels at a speed of approximately 300 000 kilometres per second. Work out an estimate of the time it takes light to travel from the Sun to the Earth. 8 An atomic particle has a lifetime of 3.86 105 seconds. It travels at a speed of 4.2 106 metres per second. Calculate an approximation for the distance it travels in its lifetime. Example 9 Use a calculator to work out a (3.4 106) (7.1 104 ) b (4.56 108) (3.2 103) a (3.4 106) (7.1 104) 2.414 1011 Use the EXP of 10x on your calculator. b (4.56 108) (3.2 103) 1.425 1011 Hint In both of these cases the brackets need not be used, but in more complex expressions the brackets must be used. 616 25.2 Using standard form Example 10 x 3.1 1012, y 4.7 1011 xy Use a calculator to work out the value of _____. xy Give your answer in standard form correct to 3 significant figures. 12 11 (3.1 10 4.7 10 ) _______________________ 12 (3.1 10 Substitute the values into the expression. 4.7 1011) 12 3.57 10 _____________ 1.457 1024 2.4502… 1012 Write the number from your calculator correctly in standard form showing more than 3 significant figures. 2.45 1012 Give your answer correct to 3 significant figures. Hint Include brackets here to ensure that the answer from the calculation on the top of the fraction is divided by the answer to the calculation on the bottom of the fraction. Exercise 25F 1 Evaluate these expressions, giving your answers in standard form. 0.08 480 a 500 600 700 b 0.006 0.004 c _________ 180 8.82 5.007 65 120 f __________ g (12.8)4 e ________ 1500 10000 2 Evaluate these expressions. Give your answers in standard form. a (3.2 1010) (6.5 106) b (1.3 107) (4.5 106) 10 6 c (2.46 10 ) (2.5 10 ) d (3.6 1020) (3.75 106) 3 Express as a number in standard form. a (3.2 108) (6.5 106) c (2.46 1010) (2.5 106) 89000 0.0086 d _____________ 48 0.25 h (36.4 24.2)3 b (1.3 107) (4.5 106) d (3.6 1020) (3.75 106) 4 Evaluate these expressions. Give your answers in standard form correct to 3 significant figures. a (3.5 1011) (6.5 106) b (1.33 1010) (4.66 104) 8 6 d (3.24 108) (6.4 104) c (5.3 10 ) (6.45 10 ) 5 Express as a number in standard form correct to 3 significant figures. a (3.5 1011) (6.5 106) b (1.33 1010) (4.66 104) d (3.24 108) (6.4 104) c (5.3 108) (6.45 106) 617 Chapter 25 Indices, standard form and surds 6 x 3.5 109, y 4.7 105 Work out the following. Give your answer in standard form correct to 3 significant figures. xy x x 2 y2 a __ b x(x 800y) c ________ d ____ y x 800y 2000 ( 7 ) x 2.4 105, y 9.6 106 Evaluate these expressions. Give your answer in standard form correct to 3 significant figures where necessary. x2 y2 xy x2 a __ b ______ c _____ y xy xy 8 The distance of the Earth from the Sun is 1.5 108 km. The distance of the planet Neptune from the Sun is 4510 million km. Write in the form 1 : n the ratio distance of the Earth from the sun : distance of the planet Neptune from the Sun 9 The mass of a uranium atom is 3.98 1022 grams. Work out the number of uranium atoms in 2.5 kilograms of uranium. Mixed exercise 25G B 1 Work out the value of these expressions. a 30 e 2 3 4 b 61 (_19 ) 1 618 5 (_52 ) 1 g d 53 (_34 ) 2 h (1 _12 ) 3 Write these numbers in standard form. a 45 000 b 0.000 62 c 894 d 0.007 21 e 100 000 000 f 0.000 000 000 007 g 90 h 0.16 Write these as ordinary numbers. a 5.8 104 b 2 105 c 4.03 107 d 4 106 e 8.45 101 f 3.152 102 g 9.2 101 h 7 104 c 89.2 104 d 5660 108 Write these in standard form. a 278 104 A f c 72 b 0.087 109 Work out and give your answer in standard form. a (5 103) (7 109) b (9 107) (2 105) c (8.4 106) (2 105) d (4.3 107) (8 103) e 800 000 0.000 000 02 f 0.000 002 4 5 000 000 25.3 Working with fractional indices 25.3 Working with fractional indices Objective Why do this? You know the meaning of fractional indices. Fractional indices are used when you model the rates at which things vibrate, such as your voice box. Get Ready Work___ out 1. √100 __ ____ 3 3. √ 27 3 2. √8 Key Points Indices can be fractions. In general, 1 __ __ an n√a In particular, this means that 1 __ 1 __ __ __ 3 a2 √a and a3 √ a Example 11 Find the value of the following 1 __ 1 __ a 252 1 __ b (1000)3 c 160.25 ___ a 25 2 √ 25 The square root of 25 is 5 because 5 5 25. 5 _______ 1 __ The cube root of 1000 is 10 because 10 10 10 1000. 3 b (1000) 3 √ 1000 10 __ c 160.25 16 1 4 1 ____ 1 __ 164 1 _____ ___ 4 √ 16 1 __ 2 Example 12 2 __ 1 Change the decimal into a fraction 0.25 __ . 4 1 __ n Use the rule a an . 1 __ Work out the value of 1 __ a 8 3 (8 3 )2 22 4 3 __ 1 b 16 4 ____ 3 __ 164 1 ______ 1 __ (164)3 1 ___ 23 1 __ 8 4 ___ 164 √16 2 because 24 16 _2 a 83 _3 b 164 Use the rule (am)n amn. Work out the cube root of 8 first. Then square your answer. 1 Use an __ . an Examiner’s Tip It is easier to work out the root first as this makes the numbers smaller and easier to manage. 619 Chapter 25 Indices, standard form and surds Exercise 25H B 1 2 3 A 4 5 Work out the value of the following. _1 _1 a 92 b 492 c 1002 Work out the value of _1 _1 a 273 b 10003 c (64)3 Work out the value of _1 _1 a 16 4 b 4 2 c 125 Work out the value of _2 _2 a 273 b 10003 c 643 6 _1 d 42 _1 _1 d 1253 _13 d _2 _23 (_15 ) 2 _13 g 8 (__321 ) _1 5 _3 d 164 Work out, as a single fraction, the value of _2 _3 _1 a 125 3 b 10 000 4 c 27 3 f 125 A _1 d 8 _23 e (_14 ) 2 e (_18 ) 3 e (_49 ) _1 _1 _1 2 _3 e 252 e 64 _32 (_25 ) Find the value of n. a _18 8n b 64 2n 2 __ 1__ c __ 5n √5 d (√7 )5 7n 3 __ e (√2 )11 2n 25.4 Using surds Objectives Why do this? You can simplify surds. You can expand expressions involving surds. You can rationalise the denominator of a fraction. 1 5 Surds occur in nature. The golden ratio _____ 2 occurs in the arrangement of branches along the stems of plants, as well as veins and nerves in animal skeletons. __ Get Ready 1. Write down the first 10 square numbers. __ ___ √ 100 2. Write down the value of a √36 b __ __ __ __ 3. Which of these has an exact answer: √5 , √ 9 , √37 , √64 ? Key Points A__number__written exactly using square roots is called a surd. and √3 are both surds. √2 __ __ 2 __ √3 and 5 √2 are examples of numbers in surd form. __ √ 4 is not a surd as √ 4 2. These two laws can be used to simplify surds. __ __ __ ___ √__ m m __ √ m √ n √ mn ___ __ √n n Simplified surds should never have a surd in the denominator. √ 620 surd √ 25.4 Using surds To rationalise the denominator of a fraction means to get rid of any surds in the denominator. __ b a__ __ . This ensures that the final fraction has an To rationalise the denominator of __ you multiply the fraction by __ √b √b integer as the denominator. √ __ √ a__ a__ __ __ ____b √b √b √b __ a__ b__ ______ √b √b √ __ a√b ____ b __ Example 13 ___ √ 12 Simplify √12 . ______ √4 3 __ __ __ ____ √ m √ n √ mn . Use __ √ 4 2. __ √4 √3 __ 2√3 Example 14 __ __ __ Expand and simplify (2 √3 )(4 √ 3 ). __ __ __ __ __ Multiply out the brackets. (2 √3 )(4 √3 ) 8 2√ 3 4√3 √ 3 √3 __ 8 6√3 3 Simplify the expression. __ 11 6√3 Exercise 25I 1 2 Find __the value of the integer k.__ __ __ a √8 k√2 b √18 k√2 c √ 50 k√ 2 Simplify ___ a √200 c √ 20 __ __ __ __ __ b √32 __ __ d √ 28 3 Solve the equation x2 30, leaving your answer in surd form. 4 Expand these expressions. Write your answers in the form a b√__c where a, b__and c are integers. __ __ __ __ a √3__(2 √ 3 ) __ b (√ 3 __1)(2 √3 ) c (√__ 5 1)(2 √5 ) 2 √ √ √ √ d ( 7 1)(2 7 ) e (2 3 ) f ( 2 5)2 5 The area of a square is 40 cm2. Find the length of one side of the square. Give your answer as a surd in its simplest form. 6 The lengths of the sides of a rectangle are (3 √ 5 ) cm and (3 √5 ) cm. Work out, in their simplified forms: a the perimeter of the rectangle b the area of the rectangle. 7 The length of the side of a square is (1__ √ 2 ) cm. Work out the area of the square. Give your answer in the form (a b√2 ) cm2 where a and b are integers. __ __ A d √ 80 k√ 5 A __ __ rationalise the denominator 621 Chapter 25 Indices, standard form and surds Example 15 __ 2__ . Rationalise the denominator of ___ √3 __ √3 2__ ___ 2__ ___ ___ __ √3 √3 √3 Multiply the fraction by ___ __ . √3 √3 __ 2 √3 ________ __ __ √3 Simplify the denominator by using __ __ the fact that √3 √3 3. √3 __ 2√3 ____ 3 __ Example 16 __ __ 15 __√ 5 and give your answer in the form a b√ 5 . Rationalise the denominator of _______ √5 __ __ √5 15 √5 ________ 15 √ 5 ___ ________ __ __ __ √5 √5 __ √5 __ Watch Out! __ Remember to multiply both parts of the expression on the top of the fraction. 15√5 √ 5 √5 ________________ __ __ √5 √5 __ 15√5 5 _________ 5 __ 1 3√5 Simplify the fraction by dividing both parts of the expression on the top of the fraction by 5. Exercise 25J A 1 2 A 3 4 5 6 622 Rationalise the denominators. 5__ 5 1__ 2__ 1__ __ b ___ c ___ d ___ e ____ a ___ √2 √7 √ 11 √3 √5 Rationalise the denominators and simplify your answers. 15__ 5 10__ 2__ 4 __ __ b ___ c ____ d ___ e ____ a ___ √2 √2 √ 12 √3 √ 10 __ Rationalise the denominators and give your answers__in the form a b√ c __where a, b and c are __integers. __ __ 12 __√3 10 __√ 5 6 __√2 14 __√ 7 2 __√2 e _______ b ______ c _______ d _______ a ______ √2 √2 √7 √3 √5 __ __ The lengths of the two shorter sides of a right-angled triangle are √ 7 cm and 2√ 3 cm. Find the length of the hypotenuse. The diagram shows a right-angled triangle. The lengths are given in centimetres. Work out the area of the triangle. __ Give your answer in the form a b√ c where a, b and c are integers. Solve these equations leaving your answers in surd form. a x2 6x 2 0 b x2 10x 14 0 2 3 9 2 Chapter review 7 C The diagram represents a right-angled triangle ABC. __ __ AB (√7 2 ) cm AC (√ 7 2 ) cm. Work out, leaving any appropriate answers in surd form: a the area of triangle ABC b the length of BC. ( 7 � 2) A ( 7 � 2) B Chapter review For non-zero values of a a0 1 For any number n 1 an __ an Standard form is used to represent very large (or very small) numbers. A number is in standard form when it is in the form a 10n where 1 a 10 and n is an integer. It is often easier to multiply and divide very large or very small numbers, or estimate a calculation, if the numbers are written in standard form. To input numbers in standard form into your calculator, use the �10 or EXP key To enter 4.5 107 press the keys 4 Indices can be fractions. In general, 1 __ · 5 �10 7 __ an n√a A number written exactly using square roots is called a surd. These two laws can be used to simplify surds. __ __ __ __ ___ √m m ___ __ √ m √ n √ mn __ √n n Simplified surds should never have a surd in the denominator. To rationalise the denominator of a fraction means to get rid of any surds in the denominator. √ __ b a__ __ , this ensures that the final fraction has an To rationalise the denominator of __ you multiply the fraction by __ √b √b integer as the denominator. √ Review exercise 1 2 3 4 5 B Work out the values of a 40 b 41 c 20 d 23 Work out the values of a 30 b (3)0 c 31 d c 2 41 2 d ___ 41 c 2 104 d 3.8 105 c 0.07 d 0.000 607 Work out the values of 1 1 a ___ b __1 31 3 Write as ordinary numbers a 3 104 b 1.67 103 ( ) Write in standard form a 5000 b 64 400 (__13 ) 0 623 Chapter 25 Indices, standard form and surds B AO2 6 a Write 150 million in standard form. The distance of the Sun from the Earth is 150 million kilometres. b Change 150 million kilometres to metres. Give your answer in standard form. 7 The number of atoms in one kilogram of helium is 1.51 1026 Calculate the number of atoms in 20 kilograms of helium. Give your answer in standard form. 8 9 10 Work out 1 __ b 1002 Work out a 90.5 b 49 Work out 1 __ a 42 AO2 AO3 1 __ a 92 11 __1 2 1 __ 1 __ c 83 c 125 June 2007 d 643 __1 3 __1 3 d 8 __1 3 b 8 Planet June 2009 Average distance from the Sun in km Mercury 5.8 107 Venus 1.1 108 Earth 1.5 108 Mars 2.3 108 Jupiter 7.8 109 Saturn 1.4 109 Uranus 2.9 109 Neptune 4.5 109 Pluto 5.9 109 The table above gives the average distance in kilometres of the nine major planets from the Sun. a Which planet is approximately 4 times further away than Mercury? b How far apart are the orbits of Neptune and Pluto? c Which planet is about half the distance from the Sun as Uranus? d Which planet is 40 times further away from the Sun than Venus? e A probe was sent from the Earth to Mars. If it took one year to reach Mars, what average speed would it have to travel? Give your answer in km/hr. 12 Estimate the value of each of the following using standard form. a 672 000 0.003 42 13 A 624 14 b (0.0543 693)2 8700 0.000 198 c ______________ 278 50 Work out (3.2 105) (4.5 104). Give your answer in standard form correct to 2 significant figures. a Write the number 40 000 000 in standard form. b Write 1.4 105 as an ordinary number. c Work out (5 105) (6 109). Give your answer in standard form. June 2005 Nov 2009 Chapter review 15 a i Write 7900 in standard form A ii Write 0.000 35 in standard form. 4 10 Give your answer in standard form. b Work out ________ 8 105 3 16 In 2003 the population of Great Britain was 6.0 107. In 2003 the population of India was 9.9 108. Work out the difference between the population of India and the population of Great Britain in 2003. Give your answer in standard form. June 2007 17 8x 2y 18 3 √27 3n 19 a √54 k√6 20 8√8 can be written in the form 8k. AO2 Express y in terms of x. __ __ Find the value of n. __ Find the value of k. June 2006 __ __ __ b √ 2 √ 8 p√ 2 Find the value of p. __ a Find the value of k. __ __ 8√8 can also be expressed in the form m√2 where m is a positive integer. b Find the value of m. 1__ c Rationalise the denominator of ___ 8√ 8 __ √2 Give your answer in the form ___ where p is a positive integer. p 21 June 2006 Work out 2 2.2 10 1.5 10 ______________________ 12 12 2.2 10 1.5 10 Give your answer in standard form correct to 3 significant figures. 12 12 Nov 2007 ______ 22 pq x _____ pq √ p 4 108 q 3 106 Find the value of x. Give your answer in standard form correct to 2 significant figures. 23 Mar 2005 ab y2 _____ ab a 3 108 b 2 107 Find y. Give your answer in standard form correct to 2 significant figures. 24 June 2003 A floppy disk can store 1 440 000 bytes of data. a Write the number 1 440 000 in standard form. A hard disk can store 2.4 109 bytes of data. b Calculate the number of floppy disks needed to store the 2.4 109 bytes of data. AO3 Nov 2003 625 Chapter 25 Indices, standard form and surds A 25 a Write 5 720 000 in standard form. p 5 720 000 q 4.5 105 pq b Find the value of _______2 (p q) Give your answer in standard form, correct to 2 significant figures. AO2 26 Winter 2005 A nanosecond is 0.000 000 001 second. a Write the number 0.000 000 001 in standard form. A computer does a calculation in 5 nanoseconds. b How many of these calculations can the computer do in 1 second? Give your answer in standard form. 27 Summer 2004 a Write 0.000 000 000 054 in standard form. S 12.6 R2 R 0.000 000 000 054 b Use the formula to calculate the value of S. Give your answer in standard form, correct to 3 significant figures. A 28 Solve 1 a 4x __ 16 29 1__ a Rationalise the denominator of __ √3 1 b 2x __ 16 __ c 2 2x __1 4 AO2 AO3 626 30 31 d 22x __1 2 __ b Expand (2 √3 ) (1 √3 ). __ Give your answer in the form a b√ 3 where a and b are integers. AO3 Winter 2005 The value of a car can be modelled by the equation: V 17 000 (0.9)t where V the value of the car in £s and t age from new in years. a Find V when t 0. b Find V when t 4. c Find the age of the car when the price first falls below £10 000. d Sketch a graph showing V against t. 1 __ ________ 1 __ ....... ________ 1 __ 1 _____ ________ __ __ Calculate ______ √2 1 √4 √3 √3 √2 10 √ 99 June 2008