Maths Age 14-16

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

2. N2 Powers, roots and standard form Contents • A N2.2 Index laws • A N2.3 Negative indices and reciprocals • A N2.1 Powers and roots N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A

3. 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.

4. Square roots squared square rooted 64 = Finding the square root is the inverse of finding the square: 8 64 We write 8 The square root of 64 is 8.

5. 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.

6. Using factors to find square roots Find 400 Find 225 If a number has factors that are square numbers then we can use these factors to find the square root. For example, √400 = √(4 × 100) √225 = √(9 × 25) = √4 × √100 = √9 × √25 = 2 × 10 = 3 × 5 = 20 = 15

7. Finding square roots of decimals Find 0.09 Find 0.0144 We can also find the square rootof a number can be made be dividing two square numbers. For example, 0.09 = (9 ÷ 100) 0.0144 = (144 ÷ 10000) = √9 ÷√100 = √144 ÷√10000 = 3 ÷ 10 = 12 ÷ 100 = 0.3 = 0.12

8. Approximate square roots key on your calculator to find out 2. Use the  If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. 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.

9. Estimating square roots What is 50? Use the key on you calculator to work out the answer.  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 50 = 7.07 (to 2 decimal places.)

10. 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, or x = 5 x = –5

11. Squares and square roots from a graph

12. 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’

13. Cube roots cubed cube rooted 125 = 5 3 Finding the cube root is the inverse of finding the cube: 5 125 We write The cube root of 125 is 5.

14. Squares, cubes and roots

15. We use index notation to show repeated multiplication by the same number. Index notation 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

16. Evaluate the following: Index notation 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. (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4× –4 × –4 × –4 = 256

17. 7 4 = xy Using a calculator to find powers We can use the xykey on a calculator to find powers. For example: to calculate the value of 74 we key in: The calculator shows this as 2401. 74 = 7 × 7 × 7 × 7 = 2401

18. N2 Powers, roots and standard form Contents N2.1 Powers and roots • A • A N2.3 Negative indices and reciprocals • A N2.2 Index laws N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A

19. When we multiply two numbers written in index form and with the same base we can see an interesting result. Multiplying numbers in index form 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 indices are added. In general, xm × xn = x(m + n) What do you notice?

20. 4 × 4 × 4 × 4 × 4 = 4 × 4 5 × 5 × 5 × 5 × 5 × 5 = 5 × 5 × 5 × 5 When we divide two numbers written in index form and with the same base we can see another interesting result. Dividing numbers in index form For example: 45 ÷ 42 = 4 × 4 × 4 = 43 = 4(5 – 2) 56 ÷ 54 = 5 × 5 = 52 = 5(6 – 4) When we divide two numbers with the same base the indices are subtracted. In general, xm÷xn=x(m – n) What do you notice?

21. 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?

22. Using index laws

23. 61 471 0.91 –51 01 The power of 1 Find the value of the following using your calculator: 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.

24. That means that: 60 = 1 Look at the following division: The power of 0 64 ÷ 64 = 1 Using the second index law, 64 ÷ 64 = 6(4 – 4) = 60 Anynon-zero number raised to the power of 0 is equal to 1. For example, 100 = 1 3.4520 = 1 723 538 5920 = 1

25. x0=1 (for x = 0) Here is a summery of the index laws you have met so far: Index laws xm × xn = x(m + n) xm÷xn=x(m – n) (xm)n=xmn x1=x

26. N2 Powers, roots and standard form Contents N2.1 Powers and roots • A N2.2 Index laws • A • A N2.3 Negative indices and reciprocals N2.4 Fractional indices • A N2.5 Surds • A N2.6 Standard form • A

27. = = 1 3 × 3 1 1 1 1 3 × 3 × 3 × 3 3 × 3 32 32 74 53 1 6 Look at the following division: Negative indices 32 ÷ 34 = Using the second index law, 32 ÷ 34 = 3(2 – 4) = 3–2 That means that 3–2 = Similarly, 6–1 = 7–4 = and 5–3 =

28. Reciprocals 1 a b The reciprocal of a is a b a The reciprocal of is x-1 We can find reciprocals on a calculator using the key. A number raised to the power of –1 gives us the reciprocalof that number. The reciprocal of a number is what we multiply the number by to get 1.

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

30. Match the reciprocal pairs

31. 1 x–1 = x 1 1 The reciprocal of x is The reciprocal of xn is x xn 1 x–n = xn Index laws for negative indices Here is a summery of the index laws for negative indices.

32. N2 Powers, roots and standard form Contents 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 • A

33. 1 1 1 1 1 1 1 1 1 1 1 1 1 9×9= 9+= 2 2 2 2 3 3 2 2 3 3 3 3 3 x = x 8 + + = 8×8 ×8 = 8 × 8 × 8 = 8 3 3 3 x = x 3 Indices can also be fractional. Suppose we have 9. Fractional indices 91 = 9 Because 3 × 3 = 9 But, 9 × 9 = 9 In general, 81 = 8 Similarly, Because 2 × 2 × 2 = 8 But, In general,

34. 3 1 3 1 What is the value of 25 ? 2 2 2 2 m n 25 We can think of as 25 . × 3 25 × 3 = (25)3 n x = (x)m Fractional indices Using the rule that (xa)b=xab we can write = (5)3 = 125 In general,

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

36. 1 2 x = x 1 m n n x = x n x = xm or (x)m n n Here is a summery of the index laws for fractional indices. Index laws for fractional indices

37. N2 Powers, roots and standard form Contents N2.1 Powers and roots • A N2.2 Index laws • A N2.3 Negative indices and reciprocals • A N2.5 Surds N2.4 Fractional indices • A • A N2.6 Standard form • A

38. 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, √9or √14 9 is not a surd because it can be written exactly.

39. 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? Like algebra, we do not use the × sign when writing surds. Using the above result, √3 × √3 × √3 = 3 × √3 = 3√3

40. 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

41. Dividing surds a b  In general, √a÷ √b = 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.

42. 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 forma√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

43. Simplifying surds Simplify the following surds by writing them in the forma√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

44. Simplifying surds

45. 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

46. Perimeter and area problem 6 1 2 3 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 = √(32 + 12) = √(9 + 1) 2√10 = √10 units √10 Length = √(62 + 22) = √(36 + 4) = √40 = 2√10 units

47. 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 + 6 1 √10 + 2√10 2√10 2 3 = 6√10 units √10 Area = √10 ×2√10 = 2 × √10 × √10 = 2 × 10 = 20 units2

48. Rationalizing the denominator Simplify the fraction 5 √2 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. 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.

49. Rationalizing the denominator Simplify the fraction ×√2 5 5 5√2 √2 = √2 ×√2 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. 2

50. Rationalizing the denominator √2 3 1) 2) 3) √5 4√7 ×√3 ×√5 ×√7 2 2√3 √2 √10 3 3√7 = = = 2 √3 √5 4√7 √3 ×√3 ×√5 ×√7 Simplify the following fractions by rationalizing their denominators. 3 5 28