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**N2 Powers, roots and standard form**

Maths Age 14-16 N2 Powers, roots and standard form

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**N2 Powers, roots and standard form**

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

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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 × or Introduce the notation 2 to mean ‘squared’ or ‘to the power of two’. Pupils sometimes confuse ‘to the power of two’ with ‘times by two’. Point out that raising to a power can be thought of as repeated multiplication. Squaring means that the number is multiplied by itself, while multiplying by two means that the number is added to itself. For example, 42 means 4 × 4 and 4 × 2 means Square numbers are given as patterns of counters in N1.1 Classifying numbers. 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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Square roots Finding the square root is the inverse of finding the square: squared 8 64 square rooted Explain that to work out the square root we need to ask ourselves ‘what number multiplied by itself will give this answer’. Ask pupils if they can think of any other number that multiplies by itself to give 64 (-8). Remind pupils that when two negative numbers are multiplied together the answer is always positive. Point out that the √ symbol refers to the positive square root by convention. Ask verbally for the square root of some known square numbers. We write 64 = 8 The square root of 64 is 8.

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**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 As an extension pupil could be asked to show this algebraically. For example, if a2 and b2 are two square numbers then a2 × b2 = aa × bb = aabb = abab = ab × ab = (ab)2. As an extension ask pupils to investigate which square numbers cannot be expressed as the product of two square numbers (the squares of prime numbers). (2 × 5)2 = 102 We can use this fact to help us find the square roots of larger square numbers.

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**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 Find 225 √400 = √(4 × 100) √225 = √(9 × 25) Ask pupils to find two square numbers that multiply together to make 400 before going through each step. Similarly, ask pupils to find two square numbers that multiply together to make 225. The general form of this rule, √ab = √a × √b, is given in N2.5 Surds. = √4 × √100 = √9 × √25 = 2 × 10 = 3 × 5 = 20 = 15

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**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, Find 0.09 Find 0.0144 0.09 = (9 ÷ 100) = (144 ÷ 10000) This rule is most useful when dealing with numbers that involve a square number divided by 100, or an even power of 10. Quotients of other square numbers are more difficult to spot. The general form of this rule, √a/b = √a ÷ √b, is given in N2.5 Surds. = √9 ÷ √100 = √144 ÷ √10000 = 3 ÷ 10 = 12 ÷ 100 = 0.3 = 0.12

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**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 This is an approximation to 9 decimal places. Explain to pupils that the square root of a number that cannot by made by multiplying or dividing two square numbers cannot be found exactly. A calculator will give an approximation to a given number of decimal places but the number cannot be written exactly as a decimal. The number of digits after the decimal point is infinite. This is called an irrational number. Ask pupils to find √2 using their calculators. Some calculators will require the √ key to be pressed after the two and some before. Make sure pupils know which way round to do this on their calculators. Stress that this is an approximation because is not 2 but a number slightly less than 2. An irrational number cannot be written exactly as a fraction or a decimal. Link: N4.2 Terminating and recurring decimals. The number of digits after the decimal point is infinite and non-repeating. This is an example of an irrational number.

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**Estimating square roots**

What is 50? 50 is not a square number but lies between 49 and 64. 50 is much closer to 49 than to 64, so 50 will be about 7.1 Therefore, 49 < 50 < 64 So, 7 < 50 < 8 Tell pupils that before using a calculator to work something out we should try to estimate the answer. To estimate the square root of a number that is not square we can estimate the answer by finding the two square numbers that the number lies between. We know that no whole number multiplies by itself to give 50 so the answer can’t be a whole number. What two square numbers does 50 lie between? Establish that the square root must lie between √49 and √64 before proceeding. Establish that if √50 lies between 7 and 8, the answer must be 7.something. Explain that since 50 is much closer to 49 than it is to 64, we would expect √50 to be closer to 7 than to 8. A good estimation therefore would be about 7.1. Tell pupils that sometimes it is better to leave √50 with the square root sign. This is an example of a surd. In N2.5 Surds we can see how √50 simplifies to 5√2. See slide 42. Use the key on you calculator to work out the answer. 50 = (to 2 decimal places.)

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**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, Before revealing –5 × –5 = 25, ask, 5 × 5 = 25, does any other number multiplied by itself give an answer of 25? Explain that the square root symbol refers to the positive square root, by convention. Pupils need to know that the negative square root exists, for example, in the solution to equations such as that shown. Ask for other negative square roots verbally. x2 = 25 has 2 solutions, x = 5 or x = –5

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**Squares and square roots from a graph**

Explain to pupils that if we plot numbers against their squares we get a graph of the shape shown on the board. This shape is called a parabola. Ask pupils to give you the coordinates of some of the points that will lie on this curve. For example, (2, 4), (–3, 9) or (1.5, 2.25). Drag the red dotted line along the y-axis to show how we can use the graph to find both the positive and negative square roots of the numbers on the y-axis. Point out that most of these numbers have been rounded to two decimal places. Unless a number can be made by multiplying or dividing square numbers, its square root cannot be written as an exact number. By dragging the the red dotted line along the x-axis, show how we can use the graph to find the squares of given numbers on the y-axis. The squares of these numbers are exact.

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**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’ Ensure that everyone understands the notation, that 43, for example, means 4 multiplied by itself three times – not 4 × 3. Tell pupils that they are expected to know the first 5 cube numbers and the answer to 10 cubed. Cube numbers are given as patterns of spheres in N1.1 Classifying numbers. 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’

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Cube roots Finding the cube root is the inverse of finding the cube: cubed 5 125 cube rooted Explain that to find the cube root we need to think ‘what number multiplied by itself and then multiplied by itself again will give this answer’. We write 125 = 5 3 The cube root of 125 is 5.

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**Squares, cubes and roots**

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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 Talk about the use of index notation as a mathematical shorthand. This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 × 2 × 2 = 32

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**Index notation Evaluate the following: 62 = 6 × 6 = 36**

When we raise a negative number to an odd power the answer is negative. 34 = 3 × 3 × 3 × 3 = 81 (–5)3 = –5 × –5 × –5 = –125 27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 When we raise a negative number to an even power the answer is positive. Talk through each example. Remind pupils that 62 can be said as ‘six squared’ or ‘six to the power of two’ and that –53 can be said as ‘negative five cubed’ or ‘negative five to the power of three’. You may wish pupils to investigate the fact that when a negative number is raised to an odd power the answer is negative and when negative numbers are raised to an even power the answer is positive. Ask pupils to justify this observation. (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4 × –4 × –4 × –4 = 256

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**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 = Reveal the steps on the board. Ensure that pupils are able to locate the power key, xy, on their calculators. Explain that we could also key in 7 × 7 × 7 × 7, but using the xy key is more efficient and we are less likely to make a mistake. State that 2401 must be a square number and ask pupils how we can show that this is true by writing 7 × 7 × 7 × 7 = (7 × 7) × (7 × 7) = 49 × 49 = 492. The calculator shows this as 2401. 74 = 7 × 7 × 7 × 7 = 2401

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**N2 Powers, roots and standard form**

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

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**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 Stress that the indices can only be added when the base is the same. = 78 = 7(3 + 5) When we multiply two numbers with the same base the indices are added. In general, xm × xn = x(m + n) What do you notice?

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**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 × 4 × 4 × 4 × 4 4 × 4 = 45 ÷ 42 = 4 × 4 × 4 = 43 = 4(5 – 2) 5 × 5 × 5 × 5 × 5 × 5 5 × 5 × 5 × 5 = 56 ÷ 54 = 5 × 5 = 52 = 5(6 – 4) Stress that the indices can only be subtracted when the base is the same. When we divide two numbers with the same base the indices are subtracted. In general, xm ÷ xn = x(m – n) What do you notice?

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**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 to a power and then raised to another power, the powers are multiplied. In general, (xm)n = xmn What do you notice?

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Using index laws Solve each problem by adding, subtracting and multiplying the indices. You may choose to include negative indices if required.

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**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.

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**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. Ask pupils to key any number into their calculator and raise it to the power of 0. The answer will always be 1. Ask them to raise 0 to the power of 0. An error message will be displayed. (00 can be considered as undefined. Pupils should not need to evaluate it. In fact, 00 is often defined as 0 or 1 according to the mathematical context. To prevent discontinuities, 00 = 1 would be preferred if you were plotting the graph of the form y = x0, for example, but 00 = 0 would be better for the graph y = 0x). For example, 100 = 1 = 1 = 1

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**xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x1 = x**

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)

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**N2.3 Negative indices and reciprocals**

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

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**Negative indices Look at the following division: 3 × 3 3 × 3 × 3 × 3 =**

1 3 × 3 = 1 32 32 ÷ 34 = Using the second index law, 32 ÷ 34 = 3(2 – 4) = 3–2 1 32 That means that 3–2 = 1 6 1 74 1 53 Similarly, 6–1 = 7–4 = and 5–3 =

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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 Tell pupils that if a number is written as a fraction we can easily find the reciprocal by swapping the numerator and the denominator. You could ask pupils to show why a/b × b/a will always equal 1. We can find reciprocals on a calculator using the key. x-1

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**Finding the reciprocals**

Find the reciprocals of the following: The reciprocal of 6 = 1 6 6-1 = 1 6 or 1) 6 2) 3 7 The reciprocal of = 7 3 or 3 7 = –1 Tell pupils that when we find the reciprocal of a decimal we can first write it as a decimal and then invert it. If we had a calculator we could also work out 1 ÷ the number or use the x--1 key. Improper fractions such as 7/3 can be written as mixed numbers if required. If a number is given as a decimal then its reciprocal is usually given as a decimal too. 0.8 = 4 5 The reciprocal of = 5 4 3) 0.8 = 1.25 or –1 = 1.25

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**Match the reciprocal pairs**

Establish that we have to find pairs of numbers that multiply together to make one. Each will be the reciprocal of the other. Point out that one number in the pair must be less than 1 and one number must be more than 1. Encourage pupils to convert decimals into fractions and mixed numbers into-top heavy fractions. The resulting fraction can then be reversed to find its reciprocal.

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**Index laws for negative indices**

Here is a summery of the index laws for negative indices. x–1 = 1 x The reciprocal of x is 1 x x–n = 1 xn Stress the relationship between negative powers and reciprocals. The reciprocal of xn is 1 xn

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**N2 Powers, roots and standard form**

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

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**Fractional indices Indices can also be fractional. Suppose we have 9 .**

9 . 1 2 9 × 9 = 1 2 = 1 2 91 = 9 But, 9 × 9 = 9 Because 3 × 3 = 9 In general, x = x 1 2 Similarly, 8 × 8 × 8 = 1 3 = 1 3 81 = 8 Elicit from pupils that the square root of a number multiplied by the square root of the same number always equals that number. We could also think of this as square rooting a number and then squaring it. Since squaring and square rooting are inverse operations this takes us back to the original number. The same is true of cube roots. If we cube root a number and then cube it we come back to the original number. Because 2 × 2 × 2 = 8 But, 8 × 8 × 8 = 8 3 In general, x = x 1 3

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**Fractional indices What is the value of 25 ? We can think of as 25 .**

3 2 We can think of as 25 3 2 1 × 3 Using the rule that (xa)b = xab we can write 1 2 25 × 3 = (25)3 = (5)3 Explain that the denominator of the power, 2, square roots the number and the numerator of the power, 3, cubes the number. = 125 In general, x = (x)m m n

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**Evaluate the following**

1) 49 1 2 49 1 2 = √49 = 7 2) 1000 2 3 1000 2 3 = (3√1000)2 = 102 = 100 1 8 3 = 1 3√8 = 1 2 3) 8- 1 3 8- 1 3 = 1 64 2 3 = 1 (3√64)2 = 1 42 = 1 16 4) 64- 2 3 64- 2 3 = 5) 4 5 2 4 5 2 = (√4)5 = 25 = 32

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**Index laws for fractional indices**

Here is a summery of the index laws for fractional indices. x = x 1 2 x = x 1 n x = xm or (x)m n m

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**N2 Powers, roots and standard form**

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

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**Which one of the following is not a surd?**

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.

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**Like algebra, we do not use the × sign when writing surds.**

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, Like algebra, we do not use the × sign when writing surds. √3 × √3 × √3 = 3 × √3 = 3√3

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**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. Pupils should notice that although √2 and √8 are irrational numbers, their product is a rational number. Similarly the product of √3 and √12 is a rational number. This is because 2 × 8 is a square number, as is 3 × 12. Use these examples to establish the general result. Ask pupils to give other examples of two surds that will multiply together to make a whole number. √3 × √12 = 6 (= √36) 6 is the square root of 36 and 3 × 12 = 36. In general, √a × √b = √ab

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** 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. Pupils should notice that although √20 and √5 are irrational numbers, their quotient is a rational number. Similarly the quotient of √18 and √2 is a rational number. This is because 20 ÷ 5 is a square number, as is 18 ÷ 2 . Ask pupils to give you examples of other surds that divide to give whole numbers. √18 ÷ √2 = 3 (= √9) 3 is the square root of 9 and 18 ÷ 2 = 9. In general, √a ÷ √b = a b

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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: Stress that when a surd is written in its simplest form, the number under the square root sign must not contain any factors that are square numbers. √50 = √(25 × 2) = √25 × √2 = 5√2

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**Simplify the following surds by writing them in the form a√b.**

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 For each example, ask pupils to give you the highest square number that divides into each surd before simplifying it. = 3√5 = 2√6 = 10√3

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Simplifying surds

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**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 Compare this to adding like terms in algebra. = 3√3 = 5√3 √27 + √75 = 3√3 + 5√3 = 8√3

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**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. Width = √( ) 1 6 = √(9 + 1) 3 2 2√10 = √10 units √10 Talk through the application of Pythagoras’ Theorem to find the width and length of the rectangle. Use the rules of surds to simplify √40 to 2√10. Length = √( ) = √(36 + 4) = √40 = 2√10 units

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**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. Perimeter = √10 + 2√10 + 1 6 √10 + 2√10 2√10 2 3 = 6√10 units √10 Verify that the area of the rectangle is 20 units squared by completing the surrounding 5 by 7 rectangle and subtracting the area of the four surrounding triangles. Point out that had we evaluated the surds we could not have been sure we had an exact answer, because we would have had to have rounded at that stage. Area = √10 × 2√10 = 2 × √10 × √10 = 2 × 10 = 20 units2

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**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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**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 = 5√2 2 ×√2

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**Rationalizing the denominator**

Simplify the following fractions by rationalizing their denominators. 1) 2 √3 2) √2 √5 3) 3 4√7 ×√3 ×√5 ×√7 2 √3 = 2√3 √2 √5 = √10 3 4√7 = 3√7 Ask pupils to suggest what the numerator and denominator of each fraction should be multiplied by before revealing the solutions. 3 5 28 ×√3 ×√5 ×√7

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**N2 Powers, roots and standard form**

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

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**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 × 10 × 10 × 10 = 104 Discuss the use of index notation to describe numbers like 10, 100 and 1000 as powers of 10. Be aware that pupils often confuse powers with multiples and reinforce the idea of a power as a number, in this case 10, repeatedly multiplied by itself. Make sure that pupils know that 103, for example, is said as “ten to the power of three”. Explain that the index tells us how many 0s will follow the 1 (this is only true for positive integer powers of ten). = 10 × 10 × 10 × 10 × 10 = 105 = 10 × 10 × 10 × 10 × 10 × 10 = 106 …

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**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 = = =10-1 1 10 101 0.01 = = = 10-2 1 102 100 0.001 = = = 10-3 1 103 1000 Talk through the use of negative integers to represent decimals. This is discussed in the context of the place value system in N4.1 Decimals and place value. = = = 10-4 1 10000 104 = = = 10-5 1 100000 105 = = = … 1 106

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**Use you calculator to work out the answer to**

Very large numbers Use you calculator to work out the answer to × Your calculator may display the answer as: 2 ×10 15 , 2 E 15 or 2 15 What does the 15 mean? Different models of calculator may show the answer in different ways. Many will leave out the ×10 and will have EXP before the power or nothing at all. Discuss how many zeros there will be in the answer. 4 × 5 is 20. There are 7 zeros in and 7 zeros in That means that the answer will have 14 zeros plus the zero from the 20, making 15 zeros altogether. The 15 means that the answer is 2 followed by 15 zeros or: 2 × 1015 =

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**Use you calculator to work out the answer to**

Very small numbers Use you calculator to work out the answer to ÷ Your calculator may display the answer as: 1.5 ×10 –12 , 1.5 E –12 or 1.5 –12 What does the –12 mean? Point out that if we include the 0 before the decimal point the answer has 12 zeros altogether. The –12 means that the 15 is divided by 1 followed by 12 zeros. 1.5 × 10-12 =

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Standard form 2 × 1015 and 1.5 × 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. Point out that the numbers between 1 and 10 do not include the number 10. 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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**Standard form – writing large numbers**

For example, the mass of the planet earth is about 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 A power of ten

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**Standard form – writing large numbers**

How can we write these numbers in standard form? = 8 × 107 = 2.3 × 108 = 7.24 × 105 Discuss how each number should be written in standard form. Notice that for large numbers the power of ten will always be one less than the number of digits in the whole part of the number. = 6.003 × 109 = × 102

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**Standard form – writing large numbers**

These numbers are written in standard form. How can they be written as ordinary numbers? 5 × 1010 = 7.1 × 106 = 4.208 × 1011 = Discuss how each number written in standard form should be written in full. 2.168 × 107 = × 103 = 6764.5

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**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 m. We write this in standard form as: 1.3 × 10-4 m. A number between 1 and 10 A negative power of 10

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**Standard form – writing small numbers**

How can we write these numbers in standard form? = 6 × 10-4 = 7.2 × 10-7 = 5.02 × 10-5 Notice that the power of ten is always minus the number of zeros before the first significant figure including the one before the decimal point.. = 3.29 × 10-8 = 1.008 × 10-3

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**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 = 9.108 × 10-8 = Again, notice that the power of ten tells us the number of zeros before the first significant figure including the one before the decimal point. 7.329 × 10-5 = × 10-2 =

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**Which number is incorrect?**

Ask pupils how the number that is incorrectly written can be expressed correctly in standard form before revealing the answer.

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**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 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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Ordering planet sizes The diameter of each planet is given in standard form. Ask a volunteer to come to the board and put the in the correct order from smallest to biggest.

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**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 Remind pupils that indices are added when we multiply. Point out that 14.4 × 108 is not in standard form and discuss how it can be converted into the correct form. This answer is not in standard form and must be converted! 14.4 × 108 = 1.44 × 10 × 108 = 1.44 × 109

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**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 Remind pupils that indices are subtracted when we divide. Discuss how 0.25 × can be converted into the correct form. This answer is not in standard form and must be converted. 0.25 × = 2.5 × 10-1 × 10-13 = 2.5 × 10-14

68
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?

69
**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 time = speed to give Time to reach Mars = 8.32 × 107 2.6 × 103 Remind pupils that 107 ÷ 103 = 104 because the indices are subtracted when dividing. = 3.2 × 104 hours This is 8.32 ÷ 2.6 This is 107 ÷ 103

70
**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. Make sure that pupils are able to enter numbers given in standard form into their calculators. Divide by 365 to give the equivalent number of years. 3.2 × 104 hours is over 3½ years.

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