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Maths Age 14-16 N2 Powers, roots and standard form 1 of 70 © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 2 of 70 © Boardworks Ltd 2008 Square numbers When we multiply a number by itself we say that we are squaring the number. To square a number we can write a small 2 after it. For example, the number 3 multiplied by itself can be written as 3×3 or 32 The value of three squared is 9. The result of any whole number multiplied by itself is called a square number. 3 of 70 © Boardworks Ltd 2008 Square roots Finding the square root is the inverse of finding the square: squared 8 64 square rooted We write 64 = 8 The square root of 64 is 8. 4 of 70 © Boardworks Ltd 2008 The product of two square numbers The product of two square numbers is always another square number. For example, 4 × 25 = 100 because 2×2×5×5=2×5×2×5 and (2 × 5)2 = 102 We can use this fact to help us find the square roots of larger square numbers. 5 of 70 © Boardworks Ltd 2008 Using factors to find square roots If a number has factors that are square numbers then we can use these factors to find the square root. For example, Find 400 6 of 70 Find 225 √400 = √(4 × 100) √225 = √(9 × 25) = √4 × √100 = √9 × √25 = 2 × 10 =3×5 = 20 = 15 © Boardworks Ltd 2008 Finding square roots of decimals We can also find the square root of a number can be made be dividing two square numbers. For example, 7 of 70 Find 0.09 Find 0.0144 0.09 = (9 ÷ 100) 0.0144 = (144 ÷ 10000) = √9 ÷ √100 = √144 ÷ √10000 = 3 ÷ 10 = 12 ÷ 100 = 0.3 = 0.12 © Boardworks Ltd 2008 Approximate square roots If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. Use the key on your calculator to find out 2. The calculator shows this as 1.414213562 This is an approximation to 9 decimal places. The number of digits after the decimal point is infinite and non-repeating. This is an example of an irrational number. 8 of 70 © Boardworks Ltd 2008 Estimating square roots What is 50? 50 is not a square number but lies between 49 and 64. Therefore, 49 < 50 < 64 So, 7 < 50 < 8 Use the 50 is much closer to 49 than to 64, so 50 will be about 7.1 key on you calculator to work out the answer. 50 = 7.07 (to 2 decimal places.) 9 of 70 © Boardworks Ltd 2008 Negative square roots 5 × 5 = 25 and –5 × –5 = 25 Therefore, the square root of 25 is 5 or –5. When we use the symbol we usually mean the positive square root. We can also write ± to mean both the positive and the negative square root. However the equation, x2 = 25 has 2 solutions, x=5 10 of 70 or x = –5 © Boardworks Ltd 2008 Squares and square roots from a graph 11 of 70 © Boardworks Ltd 2008 Cubes The numbers 1, 8, 27, 64, and 125 are all: Cube numbers 13 = 1 × 1 × 1 = 1 ‘1 cubed’ or ‘1 to the power of 3’ 23 = 2 × 2 × 2 = 8 ‘2 cubed’ or ‘2 to the power of 3’ 33 = 3 × 3 × 3 = 27 ‘3 cubed’ or ‘3 to the power of 3’ 43 = 4 × 4 × 4 = 64 ‘4 cubed’ or ‘4 to the power of 3’ 53 = 5 × 5 × 5 = 125 ‘5 cubed’ or ‘5 to the power of 3’ 12 of 70 © Boardworks Ltd 2008 Cube roots Finding the cube root is the inverse of finding the cube: cubed 5 125 cube rooted We write 125 = 5 3 The cube root of 125 is 5. 13 of 70 © Boardworks Ltd 2008 Squares, cubes and roots 14 of 70 © Boardworks Ltd 2008 Index notation We use index notation to show repeated multiplication by the same number. For example: we can use index notation to write 2 × 2 × 2 × 2 × 2 as Index or power 25 base This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 × 2 × 2 = 32 15 of 70 © Boardworks Ltd 2008 Index notation Evaluate the following: 62 = 6 × 6 = 36 34 = 3 × 3 × 3 × 3 = 81 (–5)3 = –5 × –5 × –5 = –125 When we raise a negative number to an odd power the answer is negative. 27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4 × –4 × –4 × –4 = 256 16 of 70 When we raise a negative number to an even power the answer is positive. © Boardworks Ltd 2008 Using a calculator to find powers We can use the xy key on a calculator to find powers. For example: to calculate the value of 74 we key in: 7 xy 4 = The calculator shows this as 2401. 74 = 7 × 7 × 7 × 7 = 2401 17 of 70 © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 18 of 70 © Boardworks Ltd 2008 Multiplying numbers in index form When we multiply two numbers written in index form and with the same base we can see an interesting result. For example: 34 × 32 = (3 × 3 × 3 × 3) × (3 × 3) =3×3×3×3×3×3 = 36 = 3(4 + 2) 73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7) =7×7×7×7×7×7×7×7 = 78 = 7(3 + 5) When we multiply two numbers with the same base the What do notice? indices are added. In you general, xm × xn = x(m + n) 19 of 70 © Boardworks Ltd 2008 Dividing numbers in index form When we divide two numbers written in index form and with the same base we can see another interesting result. For example: 45 56 ÷ 42 4×4×4×4×4 = = 4 × 4 × 4 = 43 = 4(5 – 2) 4×4 ÷ 54 5×5×5×5×5×5 = = 5 × 5 = 52 = 5(6 – 4) 5×5×5×5 When we divide two numbers with the same base the What do you notice? indices are subtracted. In general, xm ÷ xn = x(m – n) 20 of 70 © Boardworks Ltd 2008 Raising a power to a power Sometimes numbers can be raised to a power and the result raised to another power. For example, (43)2 = 43 × 43 = (4 × 4 × 4) × (4 × 4 × 4) = 46 = 4(3 × 2) When a number is raised todo a power and then raised to another What you notice? power, the powers are multiplied. In general, (xm)n = xmn 21 of 70 © Boardworks Ltd 2008 Using index laws 22 of 70 © Boardworks Ltd 2008 The power of 1 Find the value of the following using your calculator: 61 471 0.91 –51 01 Any number raised to the power of 1 is equal to the number itself. In general, x1 = x Because of this we don’t usually write the power when a number is raised to the power of 1. 23 of 70 © Boardworks Ltd 2008 The power of 0 Look at the following division: 64 ÷ 64 = 1 Using the second index law, 64 ÷ 64 = 6(4 – 4) = 60 That means that: 60 = 1 Any non-zero number raised to the power of 0 is equal to 1. For example, 100 = 1 24 of 70 3.4520 = 1 723 538 5920 = 1 © Boardworks Ltd 2008 Index laws Here is a summery of the index laws you have met so far: xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x1 = x x0 = 1 (for x = 0) 25 of 70 © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 26 of 70 © Boardworks Ltd 2008 Negative indices Look at the following division: 3×3 3×3×3×3 32 ÷ 34 = = 1 1 = 2 3×3 3 Using the second index law, 32 ÷ 34 = 3(2 – 4) = 3–2 That means that Similarly, 27 of 70 6–1 = 1 6 3–2 = 1 32 7–4 = 1 74 and 5–3 = 1 53 © Boardworks Ltd 2008 Reciprocals A number raised to the power of –1 gives us the reciprocal of that number. The reciprocal of a number is what we multiply the number by to get 1. 1 The reciprocal of a is a a b The reciprocal of is b a We can find reciprocals on a calculator using the x-1 key. 28 of 70 © Boardworks Ltd 2008 Finding the reciprocals Find the reciprocals of the following: 1) 6 1 The reciprocal of 6 = 6 3 2) 7 3 7 The reciprocal of = 7 3 3) 0.8 4 0.8 = 5 4 5 The reciprocal of = 5 4 or 0.8–1 = 1.25 29 of 70 or 6-1 or 3 7 1 = 6 –1 7 = 3 = 1.25 © Boardworks Ltd 2008 Match the reciprocal pairs 30 of 70 © Boardworks Ltd 2008 Index laws for negative indices Here is a summery of the index laws for negative indices. x–1 = 1 x x–n 31 of 70 = 1n x The reciprocal of x is The reciprocal of xn 1 x 1 is n x © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 32 of 70 © Boardworks Ltd 2008 Fractional indices 1 2 Indices can also be fractional. Suppose we have 9 . 1 2 1 2 9 ×9 = 9 In general, 33 of 70 1 2 = 91 = 9 Because 3 × 3 = 9 x 2 = x 1 In general, But, + 9 × 9 = 9 But, Similarly, 1 2 1 3 1 3 1 3 8 ×8 ×8 = 8 1 1 3+ 3 + 1 3 = 81 = 8 8 × 8 × 8 = 8 3 3 3 Because 2×2×2=8 x 3 = x 1 3 © Boardworks Ltd 2008 Fractional indices 3 2 What is the value of 25 ? 3 2 We can think of 25 as 25 1 2 ×3 . Using the rule that (xa)b = xab we can write 25 1 2 ×3 = (25)3 = (5)3 = 125 In general, m n n x = (x)m 34 of 70 © Boardworks Ltd 2008 Evaluate the following 49 12 = √49 = 7 1) 49 12 2) 1000 1000 23 = (3√1000)2 = 102 = 100 3) 1 8 3 1 83 4) 2 64- 3 64- 5) 4 35 of 70 2 3 5 2 1 1 1 = 1 = 3 = √8 83 2 2 3 1 1 1 1 = = 2 = 2 = 3 2 3 ( √64) 64 4 16 4 52 = (√4)5 = 25 = 32 © Boardworks Ltd 2008 Index laws for fractional indices Here is a summery of the index laws for fractional indices. x = x 1 2 n x = x 1 n m n x = 36 of 70 n xm n or (x)m © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 37 of 70 © Boardworks Ltd 2008 Surds The square roots of many numbers cannot be found exactly. For example, the value of √3 cannot be written exactly as a fraction or a decimal. The value of √3 is an irrational number. For this reason it is often better to leave the square root sign in and write the number as √3. √3 is an example of a surd. Which one of the following is not a surd? √2, √6 , √9 or √14 9 is not a surd because it can be written exactly. 38 of 70 © Boardworks Ltd 2008 Multiplying surds What is the value of √3 × √3? We can think of this as squaring the square root of three. Squaring and square rooting are inverse operations so, √3 × √3 = 3 In general, √a × √a = a What is the value of √3 × √3 × √3? Using the above result, √3 × √3 × √3 = 3 × √3 Like algebra, we do not use the × sign when writing surds. = 3√3 39 of 70 © Boardworks Ltd 2008 Multiplying surds Use a calculator to find the value of √2 × √8. What do you notice? √2 × √8 = 4 (= √16) 4 is the square root of 16 and 2 × 8 = 16. Use a calculator to find the value of √3 × √12. √3 × √12 = 6 (= √36) 6 is the square root of 36 and 3 × 12 = 36. In general, √a × √b = √ab 40 of 70 © Boardworks Ltd 2008 Dividing surds Use a calculator to find the value of √20 ÷ √5. What do you notice? √20 ÷ √5 = 2 (= √4) 2 is the square root of 4 and 20 ÷ 5 = 4. Use a calculator to find the value of √18 ÷ √2. √18 ÷ √2 = 3 (= √9) 3 is the square root of 9 and 18 ÷ 2 = 9. In general, √a ÷ √b = 41 of 70 a b © Boardworks Ltd 2008 Simplifying surds We are often required to simplify surds by writing them in the form a√b. For example, Simplify √50 by writing it in the form a√b. Start by finding the largest square number that divides into 50. This is 25. We can use this to write: √50 = √(25 × 2) = √25 × √2 = 5√2 42 of 70 © Boardworks Ltd 2008 Simplifying surds Simplify the following surds by writing them in the form a√b. 1) √45 2) √24 3) √300 √45 = √(9 × 5) √24 = √(4 × 6) √300 = √(100 × 3) = √9 × √5 = √4 × √6 = √100 × √3 = 3√5 = 2√6 = 10√3 43 of 70 © Boardworks Ltd 2008 Simplifying surds 44 of 70 © Boardworks Ltd 2008 Adding and subtracting surds Surds can be added or subtracted if the number under the square root sign is the same. For example, Simplify √27 + √75. Start by writing √27 and √75 in their simplest forms. √27 = √(9 × 3) √75 = √(25 × 3) = √9 × √3 = √25 × √3 = 3√3 = 5√3 √27 + √75 = 3√3 + 5√3 = 8√3 45 of 70 © Boardworks Ltd 2008 Perimeter and area problem The following rectangle has been drawn on a square grid. Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form. 1 6 2√10 3 √10 2 Width = √(32 + 12) = √(9 + 1) = √10 units Length = √(62 + 22) = √(36 + 4) = √40 = 2√10 units 46 of 70 © Boardworks Ltd 2008 Perimeter and area problem The following rectangle has been drawn on a square grid. Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form. 1 2√10 3 √10 Perimeter = √10 + 2√10 + 6 2 √10 + 2√10 = 6√10 units Area = √10 × 2√10 = 2 × √10 × √10 = 2 × 10 = 20 units2 47 of 70 © Boardworks Ltd 2008 Rationalizing the denominator When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number. This is called rationalizing the denominator. 5 Simplify the fraction √2 Remember, if we multiply the numerator and the denominator of a fraction by the same number the value of the fraction remains unchanged. In this example, we can multiply the numerator and the denominator by √2 to make the denominator into a whole number. 48 of 70 © Boardworks Ltd 2008 Rationalizing the denominator When a fraction has a surd as a denominator we usually rewrite it so that the denominator is a rational number. This is called rationalizing the denominator. 5 Simplify the fraction √2 ×√2 5 5√2 = 2 √2 ×√2 49 of 70 © Boardworks Ltd 2008 Rationalizing the denominator Simplify the following fractions by rationalizing their denominators. 2 1) √3 ×√3 2 2√3 = 3 √3 ×√3 50 of 70 √2 2) √5 3 3) 4√7 ×√5 √2 √10 = 5 √5 ×√5 ×√7 3 3√7 = 28 4√7 ×√7 © Boardworks Ltd 2008 Contents N2 Powers, roots and standard form A N2.1 Powers and roots A N2.2 Index laws A N2.3 Negative indices and reciprocals A N2.4 Fractional indices A N2.5 Surds A N2.6 Standard form 51 of 70 © Boardworks Ltd 2008 Powers of ten Our decimal number system is based on powers of ten. We can write powers of ten using index notation. 10 = 101 100 = 10 × 10 = 102 1000 = 10 × 10 × 10 = 103 10 000 = 10 × 10 × 10 × 10 = 104 100 000 = 10 × 10 × 10 × 10 × 10 = 105 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106 … 52 of 70 © Boardworks Ltd 2008 Negative powers of ten Any number raised to the power of 0 is 1, so 1 = 100 Decimals can be written using negative powers of ten 0.1 = 1 10 1 101 = 1 0.01 = 100 = 1 102 1 0.001 = 1000 = 0.0001 = 1 10000 0.00001 = = 10-2 1 103 = 1 100000 0.000001 = 53 of 70 =10-1 = 10-3 1 104 = 1 1000000 = 10-4 1 105 = = 10-5 1 106 = 10-6 … © Boardworks Ltd 2008 Very large numbers Use you calculator to work out the answer to 40 000 000 × 50 000 000. Your calculator may display the answer as: 2 ×1015 , 2 E 15 or 2 15 What does the 15 mean? The 15 means that the answer is 2 followed by 15 zeros or: 2 × 1015 54 of 70 = 2 000 000 000 000 000 © Boardworks Ltd 2008 Very small numbers Use you calculator to work out the answer to 0.0002 ÷ 30 000 000. Your calculator may display the answer as: 1.5 ×10–12 , 1.5 E –12 or 1.5 –12 What does the –12 mean? The –12 means that the 15 is divided by 1 followed by 12 zeros. 1.5 × 10-12 = 0.000000000002 55 of 70 © Boardworks Ltd 2008 Standard form 2 × 1015 and 1.5 × 10-12 are examples of a number written in standard form. Numbers written in standard form have two parts: A number between 1 and 10 × A power of 10 This way of writing a number is also called standard index form or scientific notation. Any number can be written using standard form, however it is usually used to write very large or very small numbers. 56 of 70 © Boardworks Ltd 2008 Standard form – writing large numbers For example, the mass of the planet earth is about 5 970 000 000 000 000 000 000 000 kg. We can write this in standard form as a number between 1 and 10 multiplied by a power of 10. 5.97 × 1024 kg A number between 1 and 10 57 of 70 A power of ten © Boardworks Ltd 2008 Standard form – writing large numbers How can we write these numbers in standard form? 80 000 000 = 8 × 107 230 000 000 = 2.3 × 108 7.24 × 105 724 000 = 6 003 000 000 = 371.45 = 58 of 70 6.003 × 109 3.7145 × 102 © Boardworks Ltd 2008 Standard form – writing large numbers These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 1010 = 50 000 000 000 7.1 × 106 = 7 100 000 4.208 × 1011 = 420 800 000 000 2.168 × 107 = 21 680 000 6.7645 × 103 = 59 of 70 6764.5 © Boardworks Ltd 2008 Standard form – writing small numbers We can write very small numbers using negative powers of ten. For example, the width of this shelled amoeba is 0.00013 m. We write this in standard form as: 1.3 × 10-4 m. A number between 1 and 10 60 of 70 A negative power of 10 © Boardworks Ltd 2008 Standard form – writing small numbers How can we write these numbers in standard form? 0.0006 = 0.00000072 = 0.0000502 = 0.0000000329 = 0.001008 = 61 of 70 6 × 10-4 7.2 × 10-7 5.02 × 10-5 3.29 × 10-8 1.008 × 10-3 © Boardworks Ltd 2008 Standard form – writing small numbers These numbers are written in standard form. How can they be written as ordinary numbers? 8 × 10-4 = 0.0008 2.6 × 10-6 = 0.0000026 9.108 × 10-8 = 0.00000009108 7.329 × 10-5 = 0.00007329 8.4542 × 10-2 = 62 of 70 0.084542 © Boardworks Ltd 2008 Which number is incorrect? 63 of 70 © Boardworks Ltd 2008 Ordering numbers in standard form Write these numbers in order from smallest to largest: 5.3 × 10-4, 6.8 × 10-5, 4.7 × 10-3, 1.5 × 10-4. To order numbers that are written in standard form start by comparing the powers of 10. Remember, 10-5 is smaller than 10-4. That means that 6.8 × 105 is the smallest number in the list. When two or more numbers have the same power of ten we can compare the number parts. 5.3 × 10-4 is larger than 1.5 × 10-4 so the correct order is: 6.8 × 10-5, 64 of 70 1.5 × 10-4, 5.3 × 10-4, 4.7 × 10-3 © Boardworks Ltd 2008 Ordering planet sizes 65 of 70 © Boardworks Ltd 2008 Calculations involving standard form What is 2 × 105 multiplied by 7.2 × 103 ? To multiply these numbers together we can multiply the number parts together and then the powers of ten together. 2 × 105 × 7.2 × 103 = (2 × 7.2) × (105 × 103) = 14.4 × 108 This answer is not in standard form and must be converted! 14.4 × 108 = 1.44 × 10 × 108 = 1.44 × 109 66 of 70 © Boardworks Ltd 2008 Calculations involving standard form What is 1.2 × 10-6 divided by 4.8 × 107 ? To divide these numbers we can divide the number parts and then divide the powers of ten. (1.2 × 10-6) ÷ (4.8 × 107) = (1.2 ÷ 4.8) × (10-6 ÷ 107) = 0.25 × 10-13 This answer is not in standard form and must be converted. 0.25 × 10-13 = 2.5 × 10-1 × 10-13 = 2.5 × 10-14 67 of 70 © Boardworks Ltd 2008 Travelling to Mars How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away? 68 of 70 © Boardworks Ltd 2008 Calculations involving standard form How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away? Rearrange speed = distance time to give time = distance speed 8.32 × 107 Time to reach Mars = 2.6 × 103 = 3.2 × 104 hours This is 8.32 ÷ 2.6 69 of 70 This is 107 ÷ 103 © Boardworks Ltd 2008 Calculations involving standard form Use your calculator to work out how long 3.2 × 104 hours is in years. You can enter 3.2 × 104 into your calculator using the EXP key: 3 . 2 EXP 4 Divide by 24 to give the equivalent number of days. Divide by 365 to give the equivalent number of years. 3.2 × 104 hours is over 3½ years. 70 of 70 © Boardworks Ltd 2008