chapter 2 indices and standard form - javedmath€¦ · chapter 2 indices and standard form 1....
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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 ( )
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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]
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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