25 INDICES, STANDARD FORM AND SURDS HI-RES STILL TO BE SUPPLIED

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1 INDICES, STNDRD FORM ND SURDS HI-RES STILL TO E SUPPLIED 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 m, or in standard form as m long. Standard form allows us to write both very large and very small numbers in a more useful form. Objectives 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. efore you start You need to be able to: use the index laws round numbers to one significant figure. 610

2 .1 Using zero and negative powers.1 Using zero and negative powers Objectives You know that n 0 1 when n 0. You know the meaning of negative indices. Why do this? If you are x metres from a live band, the volume of sound they are producing is directly proportional to x. This means that if you halve your distance to the band, the music will get four times as loud. Get Ready Work out 1... Key Points For non-zero values of a a 0 1 For any number n a n 1 a n Example 1 a 0 1 Work out the value of a 0 b 1 c 6 d ( ) ny number to the power of zero is 1. b 1 1 Use the rule a n 1 a n c d ( ) 1 ( ) ( ) 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. Examiner s Tip 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

3 Chapter Indices, standard form and surds Exercise Questions in this chapter are targeted at the grades indicated. 1 Write down the value of these expressions. a 7 0 b 8 1 c 1 d 0 e ( ) f 9 g 10 h 1 0 i ( ) j ( 8) 0 k 16 0 l 10 6 Work out the value of these expressions. a ( 1_ ) 1 b ( _ ) 1 c 7 ( 1_ ) d 7 ( 1_ ) e (0.) f ( _ ) g ( _ ) 0 h ( 9_ ) 1 i ( 1 _ ) j ( 1 1_ ) k (0.1) l (0.). Using standard form Objectives 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. Get Ready Why do this? stronomers use standard form to record large measurements. The Sun s diameter is about km. iologists working with microorganisms sometimes use standard form to record their very small sizes, like.1 10 cm. 1. Work out a 10 b 10. Write as a power of 10.. Work out Key Points Standard form is used to represent very large (or very small) numbers. number is in standard form when it is in the form a 10 n 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 number in standard form looks like this This part is written as a This part is written as a number between 1 and 10. power of 10. These numbers are all in standard form. 10, , These numbers are not in standard form , 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 press the keys standard form integer

4 . Using standard form Example Write these numbers in standard form. a b c 68. a b c Use.6 not.6 or 6 as.6 is between 1 and 10. Example Write as an ordinary number a b a b Exercise 1 Write these numbers in standard form. a b 600 c 000 d e Write these as ordinary numbers. a 6 10 b 1 10 c 8 10 d 10 8 e Write these numbers in standard form. a 000 b c 6 d.7 e 60 Write these as ordinary numbers. a b c d e In 008 there were approximately people in the world. Write this number in standard form. 6 The circumference of Earth is approximately km. Write this number in standard form. Example Write in standard form a b a b is equivalent to Using a n 1 a n Use.6 rather than 6 as.6 is between 1 and

5 Chapter Indices, standard form and surds Example Write as an ordinary number a 10 6 b a 10 6 b Exercise C 1 Write these numbers in standard form. a 0.00 b 0.0 c d 0.9 e Write these as ordinary numbers. a 6 10 b 8 10 c 10 7 d 10 1 e Write these numbers in standard form. a b c d e Write these as ordinary numbers. a b c d e Write these numbers in standard form. a b 0.00 c d e 0.78 f g 00 h i 60 j Write these as ordinary numbers. a.1 10 b 10 c d 10 6 e f g h i.7 10 j micron is of a metre. Write down the size of a micron, in metres, in standard form. 8 particle of sand has a diameter of 0.06 mm. Write this number in standard form. Example 6 Write in standard form a 0 10 b Method 1 a Write 0 in standard form Use the rule a m a n a m n. 10 b Write in standard form. Use a m a n a m n. Examiner s Tip The power of 10 tells you how many 0s there are zeros zeros 61

6 . Using standard form Method a b Work out the calculation. Change the answer into standard form. Use the rule a n 1. a n 1 Multiplying by is the same as dividing by Exercise D 1 Write these in standard form. a 10 b c d Write these in standard form. a b c d 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 b c d e f g h i j k l Write these numbers in order of size. Start with the smallest number , , , Write these numbers in order of size. Start with the smallest number ,. 10, , 7 10 Example 7 Work out ( 10 6 ) ( 10 ) giving your answer in standard form. ( 10 6 ) ( 10 ) Rearrange the expression so the powers of 10 are together. Multiply the numbers. Use a m a n a m n to multiply the powers of is not in standard form. Write your final answer in standard form. Example y writing and in standard form correct to one significant figure, work out an approximation for correct to one significant figure correct to one significant figure Rearrange the expression so the powers of 10 are together. Divide the numbers. Use a m a n a m n to divide the powers of

7 Chapter Indices, standard form and surds Exercise E 1 Work out and give your answer in standard form. a ( 10 8 ) ( 10 ) b (6 10 ) (1. 10 ) c ( 10 7 ) ( 10 ) d ( ) ( 10 6 ) e ( ) ( 10 ) f ( 10 8 ) ( 10 ) Work out and give your answer in standard form. a ( 10 8 ) ( 10 ) b (9 10 ) ( 10 ) c ( 10 9 ) (6 10 ) d ( ) ( 10 1 ) e ( ) ( 10 ) f ( ) (7 10 ) Express in standard form. a ( 10 ) b ( 10 ) c ( 10 6 ) d ( ) y 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 b c d Light travels at 10 8 metres per second. Work out the time it takes light to travel: a 00 metres b 1. centimetres. 6 The base of a microchip is in the shape of a rectangle. Its length is 10 mm and its width is mm. Find the area of the base. Give your answer in mm in standard form. O O O 7 The distance of the Earth from the Sun is approximately miles. Light travels at a speed of approximately kilometres per second. Work out an estimate of the time it takes light to travel from the Sun to the Earth. 8 n atomic particle has a lifetime of seconds. It travels at a speed of metres per second. Calculate an approximation for the distance it travels in its lifetime. Example 9 Use a calculator to work out a ( ) ( ) b ( ) (. 10 ) a ( ) ( ) Use the EXP of 10 x on your calculator. b ( ) (. 10 ) Hint In both of these cases the brackets need not be used, but in more complex expressions the brackets must be used. 616

8 . Using standard form Example 10 x , y Use a calculator to work out the value of x y xy. Give your answer in standard form correct to significant figures. ( ) ( ) Substitute the values into the expression. Write the number from your calculator correctly in standard form showing more than significant figures. Give your answer correct to 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 F 1 Evaluate these expressions, giving your answers in standard form. a b c e 6 10 f g (1.8) d h (6..) Evaluate these expressions. Give your answers in standard form. a ( ) ( ) b ( ) ( ) c ( ) ( ) d ( ) ( ) Express as a number in standard form. a ( ) ( ) b ( ) ( ) c ( ) ( ) d ( ) ( ) Evaluate these expressions. Give your answers in standard form correct to significant figures. a ( ) ( ) b ( ) ( ) c ( ) ( ) d ( ) (6. 10 ) Express as a number in standard form correct to significant figures. a ( ) ( ) b ( ) ( ) c ( ) ( ) d ( ) (6. 10 ) 617

9 Chapter Indices, standard form and surds 6 x. 10 9, y.7 10 Work out the following. Give your answer in standard form correct to significant figures. a x b x(x 800y) c xy y x 800y 7 x. 10, y Evaluate these expressions. Give your answer in standard form correct to significant figures where necessary. a x b x y c xy y x y x y 8 The distance of the Earth from the Sun is km. The distance of the planet Neptune from the Sun is 10 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 grams. Work out the number of uranium atoms in. kilograms of uranium. d ( x 000 ) y Mixed exercise G 1 Work out the value of these expressions. a 0 b 6 1 c 7 d e ( 1_ 9 ) 1 f ( _ ) 1 g ( _ ) h ( 1 1_ ) Write these numbers in standard form. a 000 b c 89 d e f g 90 h 0.16 Write these as ordinary numbers. a.8 10 b 10 c d 10 6 e f.1 10 g h 7 10 Write these in standard form. a b c d Work out and give your answer in standard form. a ( 10 ) ( ) b ( ) ( 10 ) c ( ) ( 10 ) d ( ) (8 10 ) e f

10 . Working with fractional indices. Working with fractional indices Objective You know the meaning of fractional indices. Why do this? Fractional indices are used when you model the rates at which things vibrate, such as your voice box. Get Ready Work out 1. _ Key Points Indices can be fractions. In general, a 1 n n a In particular, this means that a 1 a and a 1 a Example 11 a 1 _ Find the value of the following a 1 b ( 1000) 1 c The square root of is because. b ( 1000 ) The cube root of 1000 is 10 because c _ 16 1 Example 1 Change the decimal into a fraction 0. 1 Use the rule a n 1. a n _ 16 because 16 Work out the value of a 8 _ b 16 _. a 8 ( 8 1 ) b (16 1 ) Use the rule (a m ) n a mn. Work out the cube root of 8 first. Then square your answer. Use a n 1 a n. Examiner s Tip It is easier to work out the root first as this makes the numbers smaller and easier to manage. 619

11 Chapter Indices, standard form and surds Exercise H 1 Work out the value of the following. a 9 1_ b 9 1_ c 100 1_ d 1_ e ( 1_ ) 1_ Work out the value of a 7 1_ b _ c ( 6) 1_ d 1 1_ e ( 1_ ) 1_ 8 Work out the value of a 16 1_ b _ 1 c 1 _ 1 d ( 1 ) 1 e ( _ Work out the value of a 7 _ b 1000 _ c 6 _ d 16 _ e _ Work out, as a single fraction, the value of a 1 _ _ b c 7 _ 1 d 8 _ e 6 _ f 1 _ ( 1_ ) g 8 1_ ( _ ) 6 Find the value of n. a 1_ 8 8n b 6 n c 1 _ 1 9 ) n d ( 7 ) 7 n e ( ) 11 n. Using surds Objectives You can simplify surds. You can expand expressions involving surds. You can rationalise the denominator of a fraction. Why do this? Surds occur in nature. The golden ratio 1 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.. Write down the value of a 6 b _ 100. Which of these has an exact answer:, 9, 7, 6? Key Points number written exactly using square roots is called a surd. and are both surds. and are examples of numbers in surd form. is not a surd as. These two laws can be used to simplify surds. m n _ m mn n m n Simplified surds should never have a surd in the denominator. 60 surd

12 . Using surds To rationalise the denominator of a fraction means to get rid of any surds in the denominator. To rationalise the denominator of a you multiply the fraction by b b. This ensures that the final fraction has an b integer as the denominator. a a b b b b a b b b a b b Example 1 Simplify 1. _ 1 Example 1 Use m n mn.. Expand and simplify ( )( ). ( )( ) 8 Multiply out the brackets Simplify the expression. Exercise I 1 Find the value of the integer k. a 8 k b 18 k c 0 k d 80 k Simplify a _ 00 b c 0 d 8 Solve the equation x 0, leaving your answer in surd form. Expand these expressions. Write your answers in the form a b c where a, b and c are integers. a ( ) b ( 1)( ) c ( 1)( ) d ( 7 1)( 7 ) e ( ) f ( ) The area of a square is 0 cm. 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 ( ) cm and ( ) 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 ) cm. Work out the area of the square. Give your answer in the form (a b ) cm where a and b are integers. rationalise the denominator 61

13 Chapter Indices, standard form and surds Example 1 Rationalise the denominator of. Multiply the fraction by. Simplify the denominator by using the fact that. Rationalise the denominator of 1 Example and give your answer in the form a b. Watch Out! Remember to multiply both parts of the expression on the top of the fraction. Simplify the fraction by dividing both parts of the expression on the top of the fraction by. Exercise J 1 Rationalise the denominators. a 1 b 1 c d 7 e 11 Rationalise the denominators and simplify your answers. a 10 b 1 c d 10 e 1 Rationalise the denominators and give your answers in the form a b c where a, b and c are integers. a b 6 10 c 1 d e The lengths of the two shorter sides of a right-angled triangle are 7 cm and 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. 6 Solve these equations leaving your answers in surd form. a x 6x 0 b x 10x

14 Chapter review 7 The diagram represents a right-angled triangle C. ( 7 ) cm C ( 7 ) cm. Work out, leaving any appropriate answers in surd form: a the area of triangle C b the length of C. ( 7 ) C ( 7 ) Chapter review For non-zero values of a a 0 1 For any number n a n 1 a n Standard form is used to represent very large (or very small) numbers. number is in standard form when it is in the form a 10 n 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 press the keys 10 7 Indices can be fractions. In general, a 1 n n a number written exactly using square roots is called a surd. These two laws can be used to simplify surds. m n _ m mn n m 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. To rationalise the denominator of a you multiply the fraction by b b, this ensures that the final fraction has an b integer as the denominator. Review exercise 1 Work out the values of a 0 b 1 c 0 d Work out the values of a 0 b ( ) 0 c 1 d ( 1 ) 0 Work out the values of a 1 1 b ( 1 ) 1 c 1 d 1 Write as ordinary numbers a 10 b c 10 d.8 10 Write in standard form a 000 b 6 00 c 0.07 d

15 Chapter Indices, standard form and surds 6 a Write 10 million in standard form. The distance of the Sun from the Earth is 10 million kilometres. b Change 10 million kilometres to metres. Give your answer in standard form. O 7 The number of atoms in one kilogram of helium is Calculate the number of atoms in 0 kilograms of helium. Give your answer in standard form. June Work out a 9 1 b c 8 1 d Work out a 9 0. b 9 1 c 1 1 d Work out a 1 b 8 1 June 009 O O 11 Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto verage distance from the Sun in km The table above gives the average distance in kilometres of the nine major planets from the Sun. a Which planet is approximately 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 0 times further away from the Sun than Venus? e 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. 1 Estimate the value of each of the following using standard form. a b (0.0 69) c Work out (. 10 ) (. 10 ). Give your answer in standard form correct to significant figures. June 00 1 a Write the number in standard form. b Write as an ordinary number. c Work out ( 10 ) ( ). Give your answer in standard form. Nov 009 6

16 Chapter review 1 a i Write 7900 in standard form ii Write in standard form. b Work out 10 Give your answer in standard form In 00 the population of Great ritain was In 00 the population of India was Work out the difference between the population of India and the population of Great ritain in 00. Give your answer in standard form. June 007 O 17 8 x y Express y in terms of x n Find the value of n. June a k 6 Find the value of k. b 8 p Find the value of p can be written in the form 8 k. a Find the value of k. 8 8 can also be expressed in the form m where m is a positive integer. b Find the value of m. c Rationalise the denominator of 1 1 Work out 8 8 Give your answer in the form where p is a positive integer. June 006 p Give your answer in standard form correct to significant figures. Nov 007 p q x pq p 10 8 q 10 6 Find the value of x. Give your answer in standard form correct to significant figures. Mar 00 y ab a b a 10 8 b 10 7 Find y. Give your answer in standard form correct to significant figures. June 00 floppy disk can store bytes of data. a Write the number in standard form. hard disk can store. 109 bytes of data. b Calculate the number of floppy disks needed to store the bytes of data. Nov 00 O 6

17 Chapter Indices, standard form and surds a Write in standard form. p q. 10 p q b Find the value of (p q) Give your answer in standard form, correct to significant figures. Winter 00 O 6 nanosecond is second. a Write the number in standard form. computer does a calculation in nanoseconds. b How many of these calculations can the computer do in 1 second? Give your answer in standard form. Summer 00 7 a Write in standard form. S 1.6 R R b Use the formula to calculate the value of S. Give your answer in standard form, correct to significant figures. Winter 00 8 Solve a x 1 16 b x a Rationalise the denominator of 1 c x 1 d x 1 b Expand ( ) (1 ). Give your answer in the form a b where a and b are integers. June 008 O O O 0 The value of a car can be modelled by the equation: V (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. c Find the age of the car when the price first falls below d Sketch a graph showing V against t. 1 Calculate