Chapter 2 Indices and Standard Form

1. Simplify .)()2( 4232 yxxy (A) 10116 yx (B) 24246 yx (C) 10118 yx (D) 24248 yx (E) 24118 yx ( )

2. Simplify 3

2

2

42

23)2(

yx

xyx

÷ .

(A) 59

34 yx (B) 5

9

316

yx (C) 57

316 yx

(D) yx5

532 (E) yx5

364 ( )

3. Simplify yxxy 23 2)2( ÷ . (A) 23xy (B) 24xy (C) yx54 (D) 253 yx (E) 28xy ( ) 4. Solve the equation x32 = 16.

(A) 21 (B)

54 (C)

411 (D)

41 (E) Cannot be solved. ( )

5. Simplify 12

232 )(−

−

yxyx .

(A) 72 yx− (B) 26 yx− (C) 66 yx− (D) 76 yx− (E) 46 yx ( ) 6. Simplify 1010 32 × . (A) 105 (B) 205 (C) 106 (D) 206 (E) 1006 ( ) 7. Simplify xx −+ × 22 84 . (A) 12 (B) 212 (C) x−1012 (D) x−34 (E) x2 ( )

8. Simplify 322

4

3010

zyxxy .

(A) 3

2

3xzxy (B) 32

2

32

zxxy (C) 3

2

3xzy (D)

xyy

62 2

(E) 2

2

3xzy ( )

Page 1 of 16

9. Evaluate − (−18 )

− 23

.

(A) 4 (B) − 4 (C) 14 (D) − 14 (E) −12 ( )

10. Solve the equation 823

= (24x)

12

. (A) 16 (B) 12 (C) 23 (D) 1 12 (E) 1 34 ( )

11. Simplify 22x ÷ 43x × 6412 x

. (A) 2x (B) 2−x (C) 4x (D) 42x (E) 8x ( )

12. (x− 13

)−3 is equal to

(A) x−313

(B) x 19

(C) x (D) x0 (E) x−1 ( )

13. Find the value of −(− 127 )−

23

.

(A) 9 (B) −9 (C) 19 (D) − 19 (E) 13 ( )

14. Solve the simultaneous equations 4x + y = 16; 3x – y = 81. (A) x = 4, y = 0 (B) x = 4, y = −1 (C) x = 3, y = 1 (D) x = 3, y = 0 (E) x = 3, y = −1 ( )

Page 2 of 16

Answers 1. C 2. D 3. B 4. B 5. D 6. C 7. C 8. C 9. D 10. C 11. B 12. C 13. B 14. E

Page 3 of 16

Chapter 2 Indices and Standard Form

1. Evaluate 2 3 1 23

3 2 1+ − −( ) . [2]

2. Simplify (-2x 3 y 2 ) 4 ÷ 8x 2 y −3 . [2]

3. (a) Find the value of ( ) ( ) ( )1

234

56

2 0 2− × × [2]

(b) Simplify the following and leave your answer in positive indices only: (2x−3y 2 )−2 (3x 3 y−2) 2 [2] 4. Evaluate each of the following: (a) (8 0 )−14 + 2−1 [2]

(b) ( ) ( )23

25

58

1 2− −× ÷ [2]

5. Evaluate each of the following: (a) 4 2 × 4 × 4 0 [1]

(b) ( )14

2− [1]

(c) ( ) ( )23

34

3 0− ÷ [1]

6. Simplify (a) 2

3

3 3

3 2( )

( )−xx

[1]

(b) (2xy−2) 3 ÷ (4x−2y 3 ) 2 [2] 7. Simplify and express your answer in terms of positive indices only:

(a) ( )x y z− −8 6 432 [2]

(b) (a 2 b 3 )−3 ÷ a −4b−7 [2]

Page 4 of 16

8. Simplify the following, expressing your answer in positive index form:

(a) x y

x y

3 2

1 3

−

− [1]

(b) ( )a bab

−

−

2 3 4

4 [2]

9. Simplify the following:

(a) 2m 4 × 3m−5 ÷ m−2 [2]

(b) 35

4

54

2

78

21)2(

baab

baab

÷ [2]

10. Given that a = 2 × 103 and b = 4 × 10-5, calculate the following and leave your

answers in standard form.

(a) ab (b) ba (c) a +

b1 [6]

11. (a) Express 0.005 724 in standard form. (b) Evaluate (8 × 103) × (3.2 × 10-2), giving your answers in standard form. [4] 12. Evaluate the following and leave your answers in standard form. (a) 3.42 × 108 – 9.6 × 107 (b) (5.84 × 10-4) ÷ (2.0 × 10-15) [4] 13. Evaluate the following, giving your answers in standard form. (a) 7 (1.23 × 10-4) (b) 0.46 × 105 + 75.8 × 104 [4] 14. Given that a = 6 × 108 and b = 4 × 106, find the value of each of the following in

standard form.

(a) ba

23 (b) a – 3b [4]

15. Given that x = 2.8 × 10-5 and y = 7 × 103, find (a) 5xy2 (b) xy2 , leaving your answers

in standard form. [4] 16. (a) Rewrite 84.37 × 10-4 as a decimal. (b) Express 8.3674 × 104 in ordinary notation, correct to the nearest thousand. [3]

Page 5 of 16

17. The population of Singapore is recorded as 3 947 000 in 2005. Express 0.000 045 23 cm in standard form correct to 2 significant figures. [2]

18. The radius of a micro-organism is 0.000 045 23 cm. Express 0.000 045 23 cm in

standard form correct to 3 significant figures. [2] 19. Given that p = 9.5 × 107 and q = 5.0 × 10-6, calculate, expressing each answer in

standard form, the value of

(a) 2pq (b) qp

4 [4]

20. Evaluate each of the following: (a) (4.2 × 107) × (2.5 × 10-3) (b) (8.74 × 105) + (8.6 × 104) (c) (6.4 × 105) ÷ (20 × 10-3) [4] 21. Given that x = 6 × 103 and y = 5 × 10-4, calculate the following and leave your

answers in standard form.

(a) yx (b)

yx 2+ [4]

22. If A = 3.4 × 107 and B = 0.374 × 109, find the value of AB

A−

, giving your answers in

standard form. [4] 23. A rectangular field measures 4.5 × 102 m by 3.6 × 102 m. Calculate its

(a) area, (b) perimeter, giving your answers in standard form. [4]

24. If the area of a circle is 2.54 × 106 cm, find the

(a) radius, (b) perimeter of the circle, giving your answers in standard form correct to 3 significant figures. (Take π = 3.142) [4]

25. The population of Singapore was recently estimated to be three million, eight hundred

and eighty thousand. (a) Write the number in standard form. (b) The total land area of Singapore is approximately 640 km2. Calculate the average

number of people per square kilometer of the land area, giving your answer correct to 2 decimal places. [4]

Page 6 of 16

26. Express the following in standard form: (a) 324 kg in g, (b) 1.2 km/min in cm/s. [4]

27. Evaluate 3 6 5

4 10

4 5 6

5

× ×

×

− −

− without using a calculator. [3]

28. Simplify ( ) ( )

( )− − − −

− −

24

3 4 3 1 2

2 3 2x y xy

x y. [3]

29. Simplify a b aa b

nn

++

−÷1 3

4

2 5 . [2]

30. Simplify ( )−2

6

3 2

3x yxy

giving your answer in positive indices. [2]

31. Evaluate (a) 7 7 71

12

34

114× ÷ [2]

(b) 4 7 4

7 4

2 1 4

3 3

− −

−

× ×

× [2]

32. Evaluate each of the following, simplifying your answers as far as possible:

(a) ( ) ( ) ( )1997 11997

19971 0 0+ − [1]

(b) ( ) ( ) ( )35

82

279

04

4

3

3+ − [2]

(c) 4 5 3

15

2 7 7

7

× × [2]

33. Simplify the following: (a) (3a 4 b 7 )(5a−1b 6 ) [1]

(b) 72288

1 3

3 4m nm n

−

− [1]

34. Simplify ( ) ( )x yx y

x yx y

2 3

5 22

3 1

2 33

−

−

− −−× , giving your answer in positive indices. [3]

Page 7 of 16

35. Solve the equation 5 2 3x − = 1

25. [2]

36. Find the value of x when 6 x × 36 2 5x − = 1. [2] 37. Solve the equation 9 2 5x− =1. [2] 38. Given that 3 x = 5 and 3 y =7, find the value of 3 4 2x y− . [3] 39. Solve the equations: (a) 2 x × 4 x+2 × 8 x−1 = 64 [2] (b) 5 x ÷ 25 x−1 = 125 [2] 40. Solve the following equations: (a) 2 x × 4 x−1 = 16 [2]

(b) ( )13

9 81 2x x x÷ = + [3]

41. Find the value of x given that 4 × 3 2 1x− = 108. [2] 42. (a) Simplify 7 2 1x+ – 4 (7 2x ). [1] (b) Use the result from (a) or otherwise, and solve the equation 7 2 1x+ – 4 (7 2 x ) = 1029. [2]

43. Solve the equation 2 4 81

321 3 4x x x× ÷ =+ − . [3]

44. Evaluate (0.1)−2 × 0.22 [2]

45. (a) Evaluate (0.027)− 13

+ 160.75 + ⎝⎛

⎠⎞

12 − 1

0

+ (−3)−1. [2]

(b) Simplify (16a4)− 14

÷ (0.001a6)13

. [2] 46. (a) Evaluate

(i) ⎝⎛ ⎠⎞25681

14

(ii) ⎝⎛ ⎠⎞23

− 4 [2]

(b) Solve the equation 27x = 9 [1]

Page 8 of 16

47. Evaluate each of the following. (a) 65.5 ÷ 64.5 [1]

(b) 212

× 412

× 812

[2] 48. Evaluate

(a) 23 + (32)−15 + ⎝⎛ ⎠⎞

23

0 (b) 4

13

× 4123

[2]

49. Evaluate

(a) ⎝⎛ ⎠⎞23

− 2 (b) 8

53

(c) 5 × 532

[3]

50. Simplify 45x4y3

4z5 ÷ 15x3y5

8x2z3 . [2]

51. Given that x2y = 3, find the value of 3x6y – 9. [2] 52. Simplify each of the following.

(a) (a12

b2) 34

× (a2 b− 4)− 14

[2]

(b) 3

125x9 ÷ (81x− 4)12

[2] 53. (a) Given that 92x = 3 , find the value of x [2]

(b) Express x2 ⎝⎜⎛

⎠⎟⎞x −3

y4 −2

with positive indices. [2]

(c) Simplify (a3b−3)−2

ab and express your answer with negative indices. [2]

54. (a) Solve the equation 9x = 1

27 . [2]

(b) Simplify a54

3a4

a−3 giving your answer with positive index. [2]

Page 9 of 16

55. Evaluate

(a) 32− 45 [1]

(b) 2−3 × ⎝⎛ ⎠⎞94

−112 × ⎝⎛ ⎠⎞7

12

0 . [2]

56. Evaluate each of the following.

(a) ⎝⎛ ⎠⎞17

0 (b) (0.14)2 (c) 64−

13

[3]

57. Simplify 3a14 × 2a

− 12

12a− 2 . [2]

58. (a) Evaluate 1634 + ⎝⎛ ⎠⎞

23

−2 . [2]

(b) Given that x−3 = 4, find the value of x3. [1] 59. (a) Simplify 4x5 × 5x4. [1] (b) Find the smallest integer value of x for which 3x > 10. [2] (c) Express 2x − 3

6 − 5x − 13 + 14 as a single fraction in its lowest terms. [2]

60. Simplify the following and leave your answer in positive indices:

(a) ( )x −612 [1]

(b) ( )x y− −12 3623 [2]

61. Simplify each of the following, giving your answer in positive indices only. [8] (a) 423 −÷× xxx (b) 732 yyy ×÷ (c) 32 52 aa × (d) 433 25 aaa ÷× −

(e) aaa 4)2(6 32 ÷× (f) pqqp 4)2( 32 ÷− (g) 142 4)(2 −− ÷ qpq (h) 1432 )( −− ÷× aaa 62. Simplify each of the following, giving your answer in positive indices only. [4]

(a) 3

23

)2(48

ababa × (b) 33

22

8)2(4)3(

xyxyyxxy

÷÷

63. Simplify each of the following, giving your answer in negative indices only. [4] (a) 654 −−− ÷× xxx (b) 427 −− ×÷ aaa (c) 214 )( −−÷mm (d) 134 4)2( −− ÷ dd

Page 10 of 16

64. Simplify each of the following, giving your answer in negative indices only. [6]

(a) 5234 14)27( aaa ÷× (b) 5154 )( −−− ÷ baab

(c) 41

443

4)2(

−−

×baabba (d) 33

3

)3(28)2(3

baabba

×÷

65. Simplify each of the following, giving your answer in negative indices only. [4]

(a) 414

632

8)2()(

−− ××

aababa (b) 6522

232

)4(4)2(

yxyxyxxy

×÷−

−

66. Express each of the following as a fraction or an integer. [6] (a) 23 52 ×− (b) 430 2323 ×÷ −

(c) 212 1684 −−− ÷× (d) 222 )2()43()

211( −− −÷×

67. Express each of the following as a fraction or an integer. [12]

(a) 221 6510 −− ÷× (b) 023 )217()5(4 ÷÷ −− (c) 2122 )4()3( −− ÷

(d) 23 )3()2( −−÷− (e) 2143 )8()2( −− ÷ (f) 432 )71(497 −− ÷×

68. Solve the following equations: [4]

(a) 2433 =x (b) 6412 =x

(c) 123 =x (d) 17 −=x 69. Solve the following equations: [6]

(a) 4055 4 =x (b) 127 3 =x (c) 125

15 =x

70. Solve the following equations: [6]

(a) 362 =−x (b) 25.0823 =× x (c) x24210 =÷ 71. Simplify the following, giving your answer with positive indices. [4]

(a) ( a2

b−2 )−1 (b) (xy3)−1

(x−1y2) −3

72. Simplify the following, giving your answer with negative indices. [6]

(a) p5q6

q−3p−1 × p− 4q−5

p2q3 (b) abc−1

(a−2b)2 × a2b

(a−1c−3)−2

Page 11 of 16

73. Simplify the following expressions. [4]

(a) (2x12 ) × (6x

32 ) (b) 5x− 4

12 ÷ 4x− 1

14

(c) (2a−1)4 ÷ (8a− 12

)43

(d) 3a−2 ÷ (27a) 23

74. Evaluate the following: [3]

(a) 16932

(b) 100−

12

(c) (−8) 23

75. Evaluate the following: [6]

(a) (338 )

13

÷ ( 18 )−1 (b) (

164 )

− 13

+ (−3)−2 (c) (1 − 12 )−1 ÷ (2

14 )

− 12

76. Solve the following equations. [5]

(a) x7 = 70 (b) 5−

23

÷ 5 = 5x (c) 4x = 0.125 77. Solve the following equations. [6] (a) 82x + 1 = 32 (b) 105x − 1 = 0.001 (c) 3x − 1 × 9x + 3 = 272x − 4 78. Solve the following equations. [6]

(a) 42x − 1 = 8x + 3 (b) 1612 x + 3

= 8x + 1 79. Given that a = 4.2 × 105 and b = 8.3 × 104, find the value of the following, expressing

your answer in standard form. [4]

(a) a + b (b) a – b (c) ab (d) ab

80. Given that x-3 = 4, find the value of x3. [2]

81. If p-2 = 5 31

q , calculate the value of [4]

(a) p when q = 125, (b) q when p = 25 .

82. Given that x = 1.2 × 106, evaluate 410+x . [2]

83. Given that (ab)-2 = 21

x , find the value of x when a = 25 and b = 334 . [3]

Page 12 of 16

84. Evaluate each of the following without the use of a calculator, giving your answer in standard form correct to 4 significant figures. [10]

(a) 3.12 × 104 + 2.6 × 102 (b) 4.76× 104 − 6.13 × 103 (c) 7.91 × 109 + 6.14 × 108 (d) 3.24 × 108 − 9.86 × 107

(e) 1.02 × 10-5 + 3.19 × 10-6 (f) 8.59 × 1010 + 16.7 × 109 (g) 5.48 × 10-8 – 76.4 × 10-6 (h) 324 × 106 − 1.86 × 107 (i) 76.34 × 105 + 183.4 × 104 (j) 36.8 × 1018 − 485 × 1015 85. Use your calculator to evaluate each of the following, giving your answer in standard

form correct to 4 significant figures. [10] (a) 3.18 × 104 × 6.45 × 102 (b) 4.59× 10-3 × 8.674 × 107 (c) 5.43 × 109 ÷ (3.27 × 108) (d) 3.58 × 10-10 ÷ (7.61 × 10-9)

(e) 4.95 × 10-5 ÷ (3.14 × 10-6) (f) 6.45 × 102 ÷ (3.27 × 107) (g) 32.65 × 10-8 × 4.59 × 107 (h) 5.149 × 107 × 3.26 × 10-4 (i) 34.95 × 105 × 672.6 × 104 (j) 19.79 × 108 ÷ (39.76 × 10-3)

Page 13 of 16

Answers

1. 16 25

2. 2x10 y11 3. (a) 2 79

(b) 9

4

12

8

x

y

4. (a) 1 12

(b) 15

5. (a) 64 (b) 16 (c) 6 3

4

6. (a) −89 3x

(b) xy

7

122

7. (a) x

y z

12

9 6

(b) 12 2a b

8. (a) xy

4

5

(b) ba

12

9

9. (a) 6m

(b) ab

2

46

10. (a) 8 × 103 (b) 5 × 107 (c) 2.7 × 104

11. (a) 5.724 × 10-3 (b) 2.56 × 102

12. (a) 2.46 × 108 (b) 2.92 × 1011

13. (a) 8.61 × 10-4 (b) 8.04 × 105

14. (a) 2.25 × 102 (b) 5.88 × 108

15. (a) 6.86 × 103 (b) 5 × 108

16. (a) 0.000 843 (b) 84 000

17. 3.9 × 106

18. 4.52 × 10-5

19. (a) 9.5 × 102 (b) 4.75 × 1012

20. (a) 1.05 × 105 (b) 9.6 × 105

(c) 3.2 × 107 21. (a) 1.2 × 107 (b) 1.0 × 104

22. 1.0 × 10-1 23. (a) 1.62 × 105 m2

(b) 1.62 × 103 m24. (a) 8.99 × 102 cm2

(b) 5.65 × 103 cm 25. (a) 3.88 × 106 (b) 6.06 × 103

26. (a) 3.24 × 105 (b) 2.0 × 103

27. 1

60 28. − x

y

11

42 29. 1

2ab 30. 2

3

5xy

31. (a) 7

(b)12 14

32. (a) 1997 (b) 230 (c) 16

33. (a) 15a 3 b13

(b) nm

7

44

34. x29y 2

35. 1

2

36. 2 37. 21

2 38. 12

4937 39. (a) 1 1

6

(b) –1

40. (a) 2 (b) −1 1

7

41. x = 2 42. (a) 3(7 2x )

(b) x = 1.5 43. 3 1

6

44. 4 45. (a) 12

(b) 5a3

46. (a) (i) 1 13

(ii) 5 116

(b) 13

47. (a) 6 (b) 8

48. (a) 9 12 (b) 16

49. (a) 214

(b) 32 (c) 25

50. 6x3

z2y2

51. 72

52. (a) a− 18

b2 12

(b) 59 x5

53. (a) 18

(b) (xy)8

(c) a−7

b−5

54. (a) − 34

(b) a5712

(c) x = y − a1 + k2

55. (a) 116

(b) 127

56. (a) 1 (b) 0.0196

(c) 14

57. 12 a134

58. (a) 10 14

(b) 14

59. (a) 20x9 (b) x = 3

60. (a) 13x

(b) xy

8

24

Page 14 of 16

61. (a) 9x (b) 6y (c) 510a (d) 410a

(e) 412a (f) 7

82pq

(g) 7

4

2qp (h)

a1

62. (a) ba4 (b) 24

9xy

63. (a) 3−x (b) 51−a

(c) 10−m (d) 112 −d

64. (a) 914−a

(b) 6

15

−

−

ab (c) 218

4−− ba

(d) 18

13 −− ba

65. (a) 22131

−− ba (b) 5216 −− yx

66. (a) 813 (b) 432 (c) 2 (d) 1

67. (a) 90 (b) 6425 (c)

8116 (d) -72 (e)

641 (f) 1

68. (a) 5 (b) -6 (c) 0 (d) -1

69. (a) 3± (b) 31 (c) -3

70 (a) 61

± (b) 321− (c) 8

71. (a) 221ba

(b) 4

3

xy

72. (a) 1

q−1 (b) c−7

a−5

73. (a) 12x2 (b) 54 x−3

14

(c) a−313

(d) 13 a−1

13

74. (a) 2197 (b) 110 (c) 4

75. (a) 3

16 (b) 419 (c) 3

76. (a) 1 (b) −1 16 (c) −1

12

77. (a) − 1

12 (b) − 25 (c) 5

23

78. (a) 11 (b) 9 79. (a) 5.03 × 105 (b) 3.37 × 105 (c) 3.486 × 1010 (d) 1.976× 10-1

80. 14

81. 15

Page 15 of 16

82. 1.1 × 103

83. 1681

84. (a) 3.146 × 104 (b) 4.147 × 104 (c) 8.524 × 109 (d) 2.254 × 108

(e) 1.339 × 10-5 (f) 1.026 × 1011 (g) -7.635 × 10-5 (h) 3.054 × 108 (i) 9.468 × 106 (j) 3.632 × 1019

85. (a) 2.051 × 108 (b) 3.981 × 105 (c) 1.661 × 102 (d) 4.704 × 10-2

(e) 1.576 × 109 (f) 1.972 × 10-5 (g) 1.449 × 1013 (h) 1.679 × 104

(i) 2.351× 107 (j) 4.977 × 1010

Page 16 of 16