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© Boardworks Ltd 2008 1 of 70 N2 Powers, roots and standard form Maths Age 14-16

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© Boardworks Ltd 2008 2 of 70 A A A A A A N2.1 Powers and roots Contents N2.2 Index laws N2.4 Fractional indices N2.5 Surds N2 Powers, roots and standard form N2.3 Negative indices and reciprocals N2.6 Standard form

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© Boardworks Ltd 2008 3 of 70 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 3 2 The value of three squared is 9. The result of any whole number multiplied by itself is called a square number.

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© Boardworks Ltd 2008 4 of 70 Finding the square root is the inverse of finding the square: 8 squared 64 square rooted We write 64 = 8 The square root of 64 is 8. Square roots

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© Boardworks Ltd 2008 5 of 70 The product of two square numbers The product of two square numbers is always another square number. For example, 4 × 25 = 100 2 × 2 × 5 × 5 = 2 × 5 × 2 × 5 and (2 × 5) 2 = 10 2 We can use this fact to help us find the square roots of larger square numbers. because

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© Boardworks Ltd 2008 6 of 70 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, = 2 × 10 = 20 √ 400 = √ (4 × 100) Find 400 = 3 × 5 = 15 √ 225 = √ (9 × 25) Find 225 = √ 4 × √100= √ 9 × √25

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© Boardworks Ltd 2008 7 of 70 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, = 3 ÷ 10 = 0.3 0.09 = (9 ÷ 100) Find 0.09 = 12 ÷ 100 = 0.12 0.0144 = (144 ÷ 10000) Find 0.0144 = √ 9 ÷ √100= √ 144 ÷ √10000

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© Boardworks Ltd 2008 8 of 70 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. Approximate square roots This is an example of an irrational number.

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© Boardworks Ltd 2008 9 of 70 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 key on you calculator to work out the answer. 50 = 7.07 (to 2 decimal places.) 50 is much closer to 49 than to 64, so 50 will be about 7.1

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© Boardworks Ltd 2008 10 of 70 Negative square roots 5 × 5 = 25and–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. However the equation, x 2 = 25 has 2 solutions, x = 5 or x = –5 We can also write ± to mean both the positive and the negative square root.

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© Boardworks Ltd 2008 11 of 70 Squares and square roots from a graph

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© Boardworks Ltd 2008 12 of 70 The numbers 1, 8, 27, 64, and 125 are all: Cube numbers 1 3 = 1 × 1 × 1 = 1‘1 cubed’ or ‘1 to the power of 3’ 2 3 = 2 × 2 × 2 = 8‘2 cubed’ or ‘2 to the power of 3’ 3 3 = 3 × 3 × 3 = 27‘3 cubed’ or ‘3 to the power of 3’ 4 3 = 4 × 4 × 4 = 64‘4 cubed’ or ‘4 to the power of 3’ 5 3 = 5 × 5 × 5 = 125‘5 cubed’ or ‘5 to the power of 3’ Cubes

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© Boardworks Ltd 2008 13 of 70 Cube roots Finding the cube root is the inverse of finding the cube: 5 cubed 125 cube rooted We write The cube root of 125 is 5. 125 = 5 3

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© Boardworks Ltd 2008 14 of 70 Squares, cubes and roots

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© Boardworks Ltd 2008 15 of 70 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 2525 This number is read as ‘two to the power of five’. 2 5 =2 × 2 × 2 × 2 × 2 =32 base Index or power

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© Boardworks Ltd 2008 16 of 70 Evaluate the following: 6 2 =6 × 6 =36 3 4 =3 × 3 × 3 × 3 =81 (–5) 3 =–5 × –5 × –5 =–125 2 7 =2 × 2 × 2 × 2 × 2 × 2 × 2 =128 (–1) 5 =–1 × –1 × –1 × –1 × –1 =–1–1 (–4) 4 =–4 × –4 × –4 × –4 =256 When we raise a negative number to an odd power the answer is negative. When we raise a negative number to an even power the answer is positive. Index notation

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© Boardworks Ltd 2008 17 of 70 Using a calculator to find powers We can use the x y key on a calculator to find powers. For example: to calculate the value of 7 4 we key in: The calculator shows this as 2401. 7 4 = 7 × 7 × 7 × 7 = 2401 7 xyxy 4=

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© Boardworks Ltd 2008 18 of 70 A A A A A A N2.2 Index laws Contents N2.4 Fractional indices N2.5 Surds N2.1 Powers and roots N2.3 Negative indices and reciprocals N2.6 Standard form N2 Powers, roots and standard form

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© Boardworks Ltd 2008 19 of 70 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: 3 4 × 3 2 =(3 × 3 × 3 × 3) × (3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 3 6 7 3 × 7 5 =(7 × 7 × 7) × (7 × 7 × 7 × 7 × 7) = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 7 8 What do you notice? When we multiply two numbers with the same base the indices are added. In general, x m × x n = x ( m + n ) = 3 (4 + 2) = 7 (3 + 5)

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© Boardworks Ltd 2008 20 of 70 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: 4 5 ÷ 4 2 = 4 × 4 × 4 × 4 × 4 4 × 4 = 4 × 4 × 4 =4343 5 6 ÷ 5 4 = 5 × 5 × 5 × 5 × 5 × 5 5 × 5 × 5 × 5 = 5 × 5 =5252 What do you notice? = 4 (5 – 2) = 5 (6 – 4) When we divide two numbers with the same base the indices are subtracted. In general, x m ÷ x n = x (m – n)

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© Boardworks Ltd 2008 21 of 70 What do you notice? Raising a power to a power Sometimes numbers can be raised to a power and the result raised to another power. For example, (4 3 ) 2 =4 3 × 4 3 = (4 × 4 × 4) × (4 × 4 × 4) = 4 6 When a number is raised to a power and then raised to another power, the powers are multiplied. In general, ( x m ) n = x mn = 4 (3 × 2)

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© Boardworks Ltd 2008 22 of 70 Using index laws

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© Boardworks Ltd 2008 23 of 70 The power of 1 Any number raised to the power of 1 is equal to the number itself. In general, x 1 = x Find the value of the following using your calculator: 6161 47 1 0.9 1 –5 1 0101 Because of this we don’t usually write the power when a number is raised to the power of 1.

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© Boardworks Ltd 2008 24 of 70 The power of 0 Look at the following division: 6 4 ÷ 6 4 =1 Using the second index law, 6 4 ÷ 6 4 = 6 (4 – 4) =6060 That means that:6 0 = 1 For example, 10 0 = 13.452 0 = 1723 538 592 0 = 1 Any non-zero number raised to the power of 0 is equal to 1.

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© Boardworks Ltd 2008 25 of 70 x 0 = 1 (for x = 0) Here is a summery of the index laws you have met so far: x m × x n = x ( m + n ) x m ÷ x n = x (m – n) Index laws ( x m ) n = x mn x1 = xx1 = x x1 = xx1 = x

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© Boardworks Ltd 2008 26 of 70 A A A A A A N2.3 Negative indices and reciprocals Contents N2.4 Fractional indices N2.5 Surds N2.2 Index laws N2.1 Powers and roots N2.6 Standard form N2 Powers, roots and standard form

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© Boardworks Ltd 2008 27 of 70 Negative indices Look at the following division: 3 2 ÷ 3 4 = 3 × 3 3 × 3 × 3 × 3 = 1 3 × 3 = 1 3232 Using the second index law, 3 2 ÷ 3 4 = 3 (2 – 4) =3 –2 That means that3 –2 = 1 3232 Similarly,6 –1 = 1 6 7 –4 = 1 7474 and5 –3 = 1 5353

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© Boardworks Ltd 2008 28 of 70 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. The reciprocal of a is 1 a The reciprocal of is a b b a We can find reciprocals on a calculator using the key. x -1

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© Boardworks Ltd 2008 29 of 70 Finding the reciprocals Find the reciprocals of the following: 1) 6 The reciprocal of 6 = 1 6 6 -1 = 1 6 or 2) 3 7 The reciprocal of = 7 3 3 7 or 3 7 = 7 3 –1 3) 0.8 0.8 = 4 5 The reciprocal of = 5 4 4 5 = 1.25 or 0.8 –1 = 1.25

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© Boardworks Ltd 2008 30 of 70 Match the reciprocal pairs

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© Boardworks Ltd 2008 31 of 70 Here is a summery of the index laws for negative indices. Index laws for negative indices x –1 = 1 x x – n = 1 xnxn The reciprocal of x is 1 x The reciprocal of x n is 1 xnxn

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© Boardworks Ltd 2008 32 of 70 A A A A A A N2.4 Fractional indices Contents N2.5 Surds N2.2 Index laws N2.1 Powers and roots N2.3 Negative indices and reciprocals N2.6 Standard form N2 Powers, roots and standard form

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© Boardworks Ltd 2008 33 of 70 Indices can also be fractional. Suppose we have Fractional indices 9 × 9 = 1 2 1 2 9 + = 1 2 1 2 9 1 =9 But, 9 × 9 = 9 In general, x = x 1 2 Similarly, 8 1 =8 But, In general, 8 × 8 × 8 = 1 3 1 3 1 3 8 + + = 1 3 1 3 1 3 8 × 8 × 8 = 8 333 x = x 1 3 3 Because 3 × 3 = 9 9. 1 2 Because 2 × 2 × 2 = 8

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© Boardworks Ltd 2008 34 of 70 Fractional indices Using the rule that ( x a ) b = x ab we can write In general, x = ( x ) m m n n What is the value of 25 ? 3 2 We can think of as 25.25 3 2 1 2 × 3 1 2 25 × 3 = ( 25) 3 = (5) 3 = 125

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© Boardworks Ltd 2008 35 of 70 Evaluate the following 1) 49 1 2 2) 1000 2 3 3) 8 - 1 3 4) 64 - 2 3 5) 4 5 2 √ 49 =49 1 2 =7 ( 3 √ 1000) 2 =1000 2 3 =10 2 =100 8-8- 1 3 = 1 8 1 3 = 1 3√83√8 = 1 2 64 - 2 3 = 1 64 2 3 = 1 ( 3 √ 64) 2 = 1 4242 = 1 16 ( √ 4) 5 =4 5 2 =2 5 =32

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© Boardworks Ltd 2008 36 of 70 Here is a summery of the index laws for fractional indices. Index laws for fractional indices x = x 1 n n 1 2 x = x m or ( x ) m n m n n

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© Boardworks Ltd 2008 37 of 70 A A A A A A N2.5 Surds Contents N2.4 Fractional indices N2.2 Index laws N2.1 Powers and roots N2.3 Negative indices and reciprocals N2.6 Standard form N2 Powers, roots and standard form

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© Boardworks Ltd 2008 38 of 70 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. For this reason it is often better to leave the square root sign in and write the number as √3. Which one of the following is not a surd? √2, √6, √9 or √14 √3 is an example of a surd. The value of √3 is an irrational number. 9 is not a surd because it can be written exactly.

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© Boardworks Ltd 2008 39 of 70 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 = 3√3 Like algebra, we do not use the × sign when writing surds.

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© Boardworks Ltd 2008 40 of 70 Multiplying surds Use a calculator to find the value of √2 × √8. What do you notice? √2 × √8 = 4 4 is the square root of 16 and 2 × 8 = 16. Use a calculator to find the value of √3 × √12. √3 × √12 = 6 6 is the square root of 36 and 3 × 12 = 36. In general, √ a × √ b = √ ab (= √16) (= √36)

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© Boardworks Ltd 2008 41 of 70 Dividing surds Use a calculator to find the value of √20 ÷ √5. What do you notice? √20 ÷ √5 = 2 2 is the square root of 4 and 20 ÷ 5 = 4. Use a calculator to find the value of √18 ÷ √2. √18 ÷ √2 = 3 3 is the square root of 9 and 18 ÷ 2 = 9. In general, √ a ÷ √ b = a b (= √4) (= √9)

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© Boardworks Ltd 2008 42 of 70 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

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© Boardworks Ltd 2008 43 of 70 Simplifying surds Simplify the following surds by writing them in the form a √ b. 1) √ 45 2) √ 243) √300 √45 = √(9 × 5) = √9 × √5 = 3√5 √24 = √(4 × 6) = √4 × √6 = 2√6 √300 = √(100 × 3) = √100 × √3 = 10√3

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© Boardworks Ltd 2008 44 of 70 Simplifying surds

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© Boardworks Ltd 2008 45 of 70 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) = √9 × √3 = 3√3 √75 = √(25 × 3) = √25 × √3 = 5√3 √27 + √75 = 3√3 + 5√3 =8√3

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© Boardworks Ltd 2008 46 of 70 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 3 √ 10 6 2 2 √ 10 Width = √ (3 2 + 1 2 ) = √(9 + 1) = √10 units Length = √ (6 2 + 2 2 ) = √(36 + 4) = √40 = 2√10 units

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© Boardworks Ltd 2008 47 of 70 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 3 √ 10 6 2 2 √ 10 Perimeter = √ 10 + 2√10 + = 6√10 units √10 + 2√10 Area = √ 10 × 2√10 = 2 × √10 × √10 = 2 × 10 = 20 units 2

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© Boardworks Ltd 2008 48 of 70 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. Simplify the fraction 5 √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.

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© Boardworks Ltd 2008 49 of 70 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. Simplify the fraction 5 √2√2 5 √2 = ×√2×√2 ×√2 2 5√2

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© Boardworks Ltd 2008 50 of 70 Rationalizing the denominator 2 √3 = ×√3×√3 ×√3 3 2√3 Simplify the following fractions by rationalizing their denominators. 1) 2 √3 2) √2 √5 3) 3 4√7 √2 √5 = ×√5×√5 ×√5 5 √103 4√7 = ×√7×√7 ×√7 28 3√7

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© Boardworks Ltd 2008 51 of 70 A A A A A A N2.6 Standard form Contents N2.5 Surds N2.4 Fractional indices N2.2 Index laws N2.1 Powers and roots N2.3 Negative indices and reciprocals N2 Powers, roots and standard form

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© Boardworks Ltd 2008 52 of 70 Powers of ten Our decimal number system is based on powers of ten. We can write powers of ten using index notation. 10 = 10 1 100 = 10 × 10 = 10 2 1000 = 10 × 10 × 10 = 10 3 10 000 = 10 × 10 × 10 × 10 = 10 4 100 000 = 10 × 10 × 10 × 10 × 10 = 10 5 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 10 6 …

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© Boardworks Ltd 2008 53 of 70 Negative powers of ten Any number raised to the power of 0 is 1, so 1 = 10 0 Decimals can be written using negative powers of ten 0.01 = = = 10 -2 1 10 2 1 100 0.001 = = = 10 -3 1 10 3 1 1000 0.0001 = = = 10 -4 1 10000 1 10 4 0.00001 = = = 10 -5 1 100000 1 10 5 0.000001 = = = 10 -6 … 1 1000000 1 10 6 0.1 = = =10 -1 1 10 1 10 1

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© Boardworks Ltd 2008 54 of 70 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: What does the 15 mean? The 15 means that the answer is 2 followed by 15 zeros or: 2 × 10 15 = 2 000 000 000 000 000 2 E 15 or 2 15 2 ×10 15,

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© Boardworks Ltd 2008 55 of 70 Very small numbers Use you calculator to work out the answer to 0.0002 ÷ 30 000 000. Your calculator may display the answer as: 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 1.5 E –12 or 1.5 –12 1.5 ×10 –12,

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© Boardworks Ltd 2008 56 of 70 Standard form 2 × 10 15 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.

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© Boardworks Ltd 2008 57 of 70 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 × 10 24 kg A number between 1 and 10 A power of ten

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© Boardworks Ltd 2008 58 of 70 How can we write these numbers in standard form? 80 000 000 = 8 × 10 7 230 000 000 = 2.3 × 10 8 724 000 = 7.24 × 10 5 6 003 000 000 = 6.003 × 10 9 371.45 = 3.7145 × 10 2 Standard form – writing large numbers

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© Boardworks Ltd 2008 59 of 70 These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 10 10 = 50 000 000 000 7.1 × 10 6 = 7 100 000 4.208 × 10 11 = 420 800 000 000 2.168 × 10 7 = 21 680 000 6.7645 × 10 3 = 6764.5 Standard form – writing large numbers

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© Boardworks Ltd 2008 60 of 70 We can write very small numbers using negative powers of ten. We write this in standard form as: For example, the width of this shelled amoeba is 0.00013 m. A number between 1 and 10 A negative power of 10 Standard form – writing small numbers 1.3 × 10 -4 m.

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© Boardworks Ltd 2008 61 of 70 How can we write these numbers in standard form? 0.0006 = 6 × 10 -4 0.00000072 = 7.2 × 10 -7 0.0000502 = 5.02 × 10 -5 0.0000000329 = 3.29 × 10 -8 0.001008 = 1.008 × 10 -3 Standard form – writing small numbers

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© Boardworks Ltd 2008 62 of 70 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 = 0.084542 Standard form – writing small numbers These numbers are written in standard form. How can they be written as ordinary numbers?

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© Boardworks Ltd 2008 63 of 70 Which number is incorrect?

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© Boardworks Ltd 2008 64 of 70 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 × 10 - 5 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 -5,1.5 × 10 -4,5.3 × 10 -4,4.7 × 10 -3

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© Boardworks Ltd 2008 65 of 70 Ordering planet sizes

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© Boardworks Ltd 2008 66 of 70 Calculations involving standard form What is 2 × 10 5 multiplied by 7.2 × 10 3 ? To multiply these numbers together we can multiply the number parts together and then the powers of ten together. 2 × 10 5 × 7.2 × 10 3 =(2 × 7.2) × (10 5 × 10 3 ) = 14.4 × 10 8 This answer is not in standard form and must be converted! 14.4 × 10 8 =1.44 × 10 × 10 8 = 1.44 × 10 9

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© Boardworks Ltd 2008 67 of 70 Calculations involving standard form What is 1.2 × 10 -6 divided by 4.8 × 10 7 ? To divide these numbers we can divide the number parts and then divide the powers of ten. (1.2 × 10 -6 ) ÷ (4.8 × 10 7 ) =(1.2 ÷ 4.8) × (10 -6 ÷ 10 7 ) = 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

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© Boardworks Ltd 2008 68 of 70 Travelling to Mars How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away?

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© Boardworks Ltd 2008 69 of 70 Calculations involving standard form Time to reach Mars = 8.32 × 10 7 2.6 × 10 3 = 3.2 × 10 4 hours Rearrange speed = distance time time = distance speed to give This is 8.32 ÷ 2.6 This is 10 7 ÷ 10 3 How long would it take a space ship travelling at an average speed of 2.6 × 10 3 km/h to reach Mars 8.32 × 10 7 km away?

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© Boardworks Ltd 2008 70 of 70 Calculations involving standard form Use your calculator to work out how long 3.2 × 10 4 hours is in years. You can enter 3.2 × 10 4 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 × 10 4 hours is over 3½ years.

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