AXIS EDUCATION Algebra [Surd/Indices] AL 1
Institute for O/A levels Teacher: Shakib Ahmed Pure Mathematics
Simplify:
1. 28 2. 72 3. 50 4. 32
12 27
5. 90 6. 7. 8. 20 80
2 3
9. 200 18 72 10. 175 63 +2 28
Rationalise the denominators:
1 1 1 3
11. 12. 13. 14.
5 11 2 15
12 5 12 7
15. 16. 17. 18.
48 80 156 63
1 1
19. 20.
1 3 2 5
1
21
21. (a) Express in the form k 7 . (b) Express 8 3 as an exact fraction in its simples form..
7
22. f (x) ( x 3)2 (1 3 x )2
Show that f(x) can be written in the form ax b, where a and b are integers to be found.
1
4
23. (a) Evaluate (5 ) 2
9
1 x
(b) Find the value of x such that 3 , giving your answer in the form a b 3 , where a and b are rational.
x
24. Express 50 3 8 in the form k 2 .
18
25. (a) Express in the form k 3 (b) Express (1 3)(4 2 3) in the form a b 3 where a and b are integers
3
2
26. Express in the form a b 5 where a and b are rational.
3 5 7
3 1
1
27. (a) solve the equation x 2 27 (b) Express (2 ) 2 as an exact fraction in is simplest form.
4
28. (a) Express each of the following in the form p q 2 , where p and q are rational.
3
1
(i) (4 3 2)2 (ii) (b) (i) Solve the equation y2 8 9y (ii) Hence solve the equation x 3 8 9 x 2
2 2
29. Find the integer n such that 4 12 75 n
1 1 1 1
30. (a) Evaluate (36 2 16 4 )3 (b) Solve the equation 3x 2 4 0
31. Express 22.5 in the form k 10 G
F
32.
B C
A D E
H
The diagram shows the rectangles ABCD and EFGH which are similar. Given that AB (3 5) cm, AD 5 cm
and EF (1 5) cm, find the length EH in cm, giving your answer in the form a b 5 , where a and b are integers.
1 2
33. Evaluate 49 2 83
34 (a) Express 80 in the form a 5 , where a is an integer.
(b) Express (4 5)2 in the form b c 5 , where b and c are integers.
1 3
35. (a) Write down the value of 16 2 . (b) Hence find the value of 16 2
36. (a) Express (5 2) in the form a b 2 , where a and b are integers.
2
(b) Hence, or otherwise, simplify (5 2)2 (5 2)2 .
1 3
37. (a) Write down the value of 16 2 (b) Find the value of 16 2
1 2
38. (a) Write down the value of 8 3 (b) Find the value of 8 3
39. Factories completely x 4 x 3x
3 2
2(3 5)
30. (a) write 45 in the form a 5 , where a is an integer (b) Express in the form b c 5 , where b
(3 5)
and c are integers.
26
41. (a) Expand and simplify (4 3)(4 3) . (b) Express in the form a b 3 , where a and b are integers.
4 3
42. (a) Express 108 in the form a 3 , where a is an integer. (b) Express (2 3)2 in the form b c 3 , where b
and c are integers to be found.
43. Simplify (3 5)(3 5) .
4
4
15 x 3
44. (a) Find the value of 83 . (b) Simplify .
3x
1 3
12 4
45. (a) Write down the value of 16 4 . (b) Simplify (16 x ) .
5 3
46. Simplify , giving your answer in the form a b 3 , where a and b are integers.
2 3
47. Factories completely x 3 9 x .
1 1
48. (a) Write down the value of 125 3 . (b) Find the value of 125 3
49. Expand and simplify ( 7 2)( 7 2) .
50. Simplify (a) (3 7)2 (b) (8 5)(2 5)
51. Given that 32 2 2 , find the value of a.
a
7 5
52. (a) Expand and simplify (7 5)(3 5) . (b) Express in the form a b 5 , where a and b are integers.
3 5
53. Write 75 27 in the form k x , where k and x are integers.
1 1
54. (a) Find the value of 16 4 (b) Simplify x(2x 4 )4 .
52 3
55. Simplify giving your answer in the form p q 3 , where p and q are rational numbers.
3 1
1 3
56. Find the value of (a) 252 (b) 25 2
57. (a) Simplify 32 18 , giving your answer in the form a 2 , where a is an integer.
32 18
(b) Simplify , giving your answer in the form b 2 c , where b and c are integers.
3 2
3 1
25 x 4 2
58. (a) Evaluate (32)5 , giving your answer as an integer. (b) Simplify fully ( )
4
2
59. Show that can be written in the form a b , where a and b are integers.
(12) (8)
60. Factories completely x 4 x 3
61. Express 82 x 3 in the form 2y , stating y in term of x.
62. (i) Express (5 8)(1 2) in the form a b 2 , where a and b are integers.
30
(ii) Express 80 in the form c 5 , where c is an integer.
5
7 5
63. Simplify , giving your answer in the form a b 5 ,where a and b are integers.
5 1
1
5
(2 x 2 )3
64. (a) Find the value of 83 (b) Simplify fully
4 x2
5 7
65. Simplify fully . (a) (2 x )2 (b)
2 7
66. Answer this question without the uses of a calculator and show all your working
4 6
(i) Show that 2 2(2 3) (ii) Show that 27 21 7 8 3
2 2 6 3
1 2
67. (a) Write down the value of 32 5 (b) Simplify fully (32x 5 ) 5
68. (a) Write 80 in the form c 5 , where c is a positive constant. A rectangle R has a length of (1 5) cm and an
area of 80cm2 . (b) Calculate the width of R in cm. Express your answer in the form p q 5 , where p and q
are integers to be found
6x
69. Without using your calculator, solve x 27 21 Write your answer in the form a b where a and b are
3
integers. You must show all stages of your working.
1
32
70. Simplify the following expressions fully. (a) 6 3
(x ) (b) 2(x 3 )
x2
71. Answer this question without the use of a calculator and show your method clearly.
20
(i) Show that 45 6 30 5 5
5
17 2
(ii) Show that 3 2 1
2 6
42 3 ab 3
72. Given that can be written in the form where a and b are integers and c is prime, find the value
52 3 c
of a ,the value of b and the value of c. [May19/P1/Q2]
Answers
1. 2 7 2. 6 2 3. 5 2 4. 4 2 5. 3 10 6. 3 7. 3
5 11 2 5
8. 6 5 9.7 2 10.12 7 11. 12. 13. 14.
5 11 2 5
1 1 13 1 1 3 2 5
15. 16. 17. 18. 19. 20.
2 4 13 3 2 1
1 3 1 1
21. (a) 3 7 (b) 22. A=10, b=10 23. (a) (b) 3 24. 11 2 25. (a) 6 3
2 7 2 2
7 3 2 1
(b) 10 6 3 26. 5 27. (a) 9 (b) 28. (a) (i) 34 24 2 (ii) 1 2 (b)
2 2 3 2
9 3
(i) y 1 , 8 (ii) x=1, 4 29. n=27 30. (a) 2 (b) 31. 10 32. 5 2 5 33. 11
16 2
1
34.(a) 4 5 (b) 21 8 5 35. (a) 4 (b) 64 36. (a) 27 10 2 (b) 20 2 37. (a) 4 (b)
64
1
38. (a) 2 (b) 39. x(x 1)(x 3) 40. (a) 3 5 (b) 7 3 5 41. (a) 13 (b) 8 2 3
4
1
42. (a) 6 3 (b) 7 4 3, b 7, c 4 43. 4 44. (a) 16 (b) 5x 3
1
45. (a) 2 (b) 8x 9 46. 13 7 3 47. x(x 3)(x 3) 48. (a) 5 (b)
5
1
49. 3 50. (a) 63 (b) 11 6 5 52. (a) 16 4 5
51. 5 (b) 4 5 53. 2 3
2
1 1 3 1
54. (a) (b) 16 55. 3 56. (a) 5 (b) 57. (a) 7 2 (b)
2 2 2 125
2
3 2 2 58. (a) 8 (b) 59. 3 2 60. x(1 2x)(1 2x) 61. y 6 x 9 62. (i) 1 3 2 (ii) 10 5
5x 2
2 1
63. 3 2 5 64. (a) 32 (b) 65. (a) 4x (b) 1 7 67. (a) 2 (b) 68. (a) 4 5
x 4x 2
1 4
(b) 5 5 69. 7 3 70. (a) x 2 (b) x 72. a=32, b=18, c=13
4
