Download Elementary Set

Chapter 1 Laws of Indices
Chapter 1
1.1
Laws of Indices
WARM-UP EXERCISE
1. Evaluate the following expressions.
(a) 6 3  4 2  3 2
(b) 12 2  (2 3  4 3)
2. Evaluate the following expressions.
(a)
35
9
4
(b) ( ) 4
6
3. Express the following numbers as a product of prime factors using the index notation.
(a) 360
(b) 945
4. Fill in the blanks with suitable numbers.
(a) 4 (
)
 256
(b) 6 (
)
 216
5. Evaluate the following expressions.
(a) (3) 3  3 3
(b) (5) 6  5 6
6. Express the following expressions in index notation.
(a) 2  5  3  3
(b) x  y  (x)  z  (z)  (y)  x
BUILD-UP EXERCISE
Exercise 1A
 Elementary Set
Level 1

1. Simplify the following expressions.
(a) a 2  a 3
(c) b 6  b 3
(b) a12  a 5  a 2
b2
(d)
10b 3
Ex.1A Elementary Set
[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Set and
Advanced Set. You may choose to complete any ONE set according to your need. ]
1.2
New Trend Mathematics S3A — Junior Form Supplementary Exercises
2. Simplify the following expressions.
(b) (  p 5 ) 5
(a) ( p 2 ) 3
(c)  (
p2 6
)
2
(d) ( 4 p 5 ) 2
3. Simplify the following expressions.
x3
(a) (xy 2 ) 2
(b) (
(c) 5( x 5 y 3 ) 2
(d) (
y2
)4
y
2x
2
)3
4. Simplify the following expressions.
(a) (c) 2 (c) 3
(b) (2d ) 2 (3d ) 3
(c) (2c) 3  (2c) 5
(d) (4 f ) 3  (2 f )
5. Given that n is a positive integer, simplify the following expressions.
Ex.1A Elementary Set
(a) 2a 5  a n  a
(b) b5n  b n
(c) ( p 3 ) n
(d) (3q n ) 4
Level 2
6. Simplify the following expressions.
(a) q 2  q 4  q 5
(b) q 6  q13  q10
(c)
3q 2  q 8
12 q 6
7. Simplify the following expressions.
(a) (
q4
q
2
)3
(b) (
3q
6q
4
)2
q
(c) 100 (
5q
6
)3
8. Simplify the following expressions.
(a) (
3x 2 2
)
xy
(b) (
4x3 y
8y
2
)4
(c) (
3x 2 y 4
2
)3
x y
9. Simplify the following expressions.
(a)
(2a 2  3b 3 ) 2
(6 a 4 ) 3
(b)
(3a 2 c) 3
(6ab) 2
10. Simplify the following expressions.
x
x
x3
x4
(a) ( 3 ) 2 ( 3 2 )
(b) ( 4 ) 2  ( 2 )
y
x y
y
y
(c)
(3a 2 ) 4 (2b 3 ) 2
(c) (
a 8b 5
4x
3y
)2  (
2
2x3
y
3
)3
Chapter 1 Laws of Indices
1.3
11. Given that n is a positive integer, simplify the following expressions.
2 n1
(b)
2n
5n
9n
(c)
5 2 n 1
3n
12. Given that n is a positive integer, simplify the following expressions.
(a)
(2 n1 ) 2
2 2n1
(b)
12  5 n3
7  2 n1  2  2 n1
(c)
20  5 n
13. Given that m is a positive integer where m  1, simplify
( x m y m1 ) 2
x m1 y m2
 Advanced Set
Level 1
2 2( n1)
Ex.1A Elementary Set
(a)
.

1. Simplify the following expressions.
(a) 2a 3  4a 2
a6
(c)
2a 4
(b) b 4  3b5  6b10
6b 2
(d)
9b
2. Simplify the following expressions.
1
(b) (
10 p
(c) ( 3 p )
2 2
2
)2
(d)  (2 p 3 ) 5
3. Simplify the following expressions.
(a) (4 x 2 y 3 ) 4
(c) (
4x
2y
)5
2
4. Simplify the following expressions.
1
(a) (3d ) 3 (2d ) 4 ( d ) 2
2
(c) (4d ) 3  (2d ) 6
(b) 2( x 3 y ) 3
(d) (
6y2
9x
3
)2
1
(b) ( d ) 4 (3d 2 )(9d 5 )
3
1
(d) (3d ) 6  ( d 2 )  (12d ) 3
2
5. Given that n is a positive integer where n  2 , simplify the following expressions.
(a) 2 p 2  3 p n 4
(b) 5 p 3n  ( p 2 )  (2 p n3 )
(c) (2 p 3n )  (4 p 2 n )
(d) (9 p 3 n )  (3 p n2 )
(e) 16( p 5 ) n
(f) ( 8 p 2 n1 ) 2
Ex.1A Advanced Set
(a) ( p 5 ) 7
1.4
New Trend Mathematics S3A — Junior Form Supplementary Exercises
Level 2
6. Simplify the following expressions.
(a) 4q 2  q 4  64 q 3
(c)
(b) 11q 5  q 2  5q 7
q4  q2
(d)
49 q15
7. Simplify the following expressions.
6q
(a) ( 3 ) 2
4q
4q
(c) 10( 4 ) 2
8q
24 q 7
(3q 2 )(6q 2 )
(b) 25(
q2
6
)3
q
3q 5
(d) 16( 2 ) 3
2q
8. Simplify the following expressions.
(a) (
4x5 y
2y
Ex.1A Advanced Set
(c) 54(
2
)2
16 xy
(b) (
5
9x 2 y
)2
2
(d) (
23 x 3 y 4
20 x y
2
)3
4 2 x 4 y 3 5
)
8 xy
9. Simplify the following expressions.
(a)
(c)
( p 2 ) 3 (3 p 4 ) 2
(b)
( p 3  9q 2 ) 2
(5 p 3 ) 2 (3q 2 ) 4
(d)
50 p 7 q 4 r
(4 pq 2 r 3 ) 2
(3 p 3 q 2 ) 3
( pq) 2 (3 p 3 qr) 2
p4q2r 5
10. Simplify the following expressions.
(a) (
(c) (
x3
x2
y
4
)3 (
2
12
9y
2
x y
)3  (
(b) (
)
3
2
32 x y
48 xy
3
)4
(d) (
x2
xy3
y
2
)2  (
4
4 xy
5y
2
)2[
)3
x y
10 xy
(2 x) 2
]
11. Simplify the following expressions.
(a)
(c)
(ab)(2a 2 b)(3a 3b 2 ) 2
18a 4 b 7
(3uv)(u 2 v) 2 (uv2 ) 3
(2u 2 v 2 ) 4
(b)
(d)
(3x 4 )(5 xy) 2 (2 x 2 y) 3
(5 x 3 y 3 ) 2
(5 p 2 q ) 4 (2 p 3 q ) 2
3
2
( 4qp ) 3 ( 5qp ) 3
Chapter 1 Laws of Indices
1.5
12. Given that n is a positive integer where n  1 , simplify the following expressions.
(a)
3n2
3 n 1
(b)
16 2 n
4 2n
13. Given that n is a positive integer, simplify the following expressions.
(c)
16  3 2 n1
24  3n
4  3n 2  5  3n1
10  3n  8  3n1
(b)
(d)
(2 n1 ) 2 (2 2 )
2 2n1
3n1 (32 ) n
9 3n1
14. Given that n is a positive integer where n  2 , simplify the following expressions.
(a)
(c)
(a n b n1 ) 2
a n1b n 2
x 2 n1  x 2 n 2
x 2 n3  x 2 n1
(b)
(a 2 b 3 ) n (ab) n1
(d)
Ex.1A Advanced Set
(a)
a n2 b n1
2 x n 3  x 2 n
x 2 n 3  2 x n  6
15. Arrange the following numbers in descending order.
1
1
1
1
(b) ( ) 40 , ( ) 61 , ( ) 32 , 123
8
4
16
2
Exercise 1B
 Elementary Set
Level 1
1. Evaluate the following expressions.
1
(a) ( ) 0
2
1
(c)  0
10

(b) 2(30 )
(d) (1) 0  1
2. Evaluate the following expressions.
(a) 4 3
(c) ( 2) 5
3. Evaluate the following expressions.
1
(a) 3
2
(c) 10(3 3 )
(b) (3 2 )
1
(d) ( ) 4
2
5 2
6
1
(d) 4( ) 2
7
(b)
Ex.1B Elementary Set
(a) 4 100, 8 68, 16 49, 32 41
1.6
New Trend Mathematics S3A — Junior Form Supplementary Exercises
4. Evaluate the following expressions.
(a) 52  34
(c)
3
1
4 2
(b)
22
3 1
10 2
(d)
10
5. Evaluate the following expressions.
(a) (53 ) 1
1
(c)
(12 1 ) 2
(b) (6 1 ) 3
1
(d) ( 2 ) 2
4
6. Evaluate the following expressions.
(a)
32 12 2
(c) (
23
23
4
1
) 2
(b) 3 4 
43
62
(d) (33  6 2 ) 1  12
7. Simplify the following expressions.
Ex.1B Elementary Set
(a) a 0
(b) 10a 0
(c) (2a) 0
(d) (5a 0 ) 2
8. Simplify the following expressions, and express your answers in positive indices.
1
(a) x 10
(b) 5
x
(c) 3( x 2 ) 3
(d) ( 4 x 5 ) 2
9. Simplify the following expressions, and express your answers in positive indices.
(a) a 2  a 3
(c) a 5  a 6
(b) a 3  a 3
a 9
(d) 1
a
10. Simplify the following expressions, and express your answers in positive indices.
(a) ( x 2 y ) 5
(b) ( x 2 y 3 ) 1
(c) (3 x 1 y 1 ) 2
(d) (
2x 2
y3
) 3
Level 2
11. Simplify the following expressions, and express your answers in positive indices.
2
(a) (a 2 b 5 )(a 4 b 6 )
(b) (3a 1b 4 )( ab3 )
3
a 3b 1
3a 2 b
(c)
(d)
2ba 3
9ab 1
Chapter 1 Laws of Indices
1.7
12. Simplify the following expressions, and express your answers in positive indices.
(c) (
2ab 3
2
a b
) 4
4
(b) [(3a 5b 2 )(2a 2 b 3 )]3
16a 5b 7
(d) [
2
2
]2
(2a b)(ab )
13. Simplify the following expressions, and express your answers in positive indices.
(a)
(a 1 ) 2 (b 3 ) 1
a 1b 2 c 4
(b)
(a 5c 3 ) 0 (4a 2b) 1
[6(ac) 1 b 2 ]2
Ex.1B Elementary Set
(a) [(b 5 c 6 ) 0 b 3 c]2
14. Which number, 25 16 , 5 33 and 125 12, is the greatest?
 Advanced Set
Level 1

1. Evaluate the following expressions.
(a) (30 ) 4
(b) [(5) 4  11]0
(c) 52  30
(d)
(6) 0
32
(a) 2 4
(b) (3) 3
(c) 5 3
(d) (4) 2
3. Evaluate the following expressions.
(a) 0.4 1
1
(c) ( ) 2
4
5
(b) ( ) 3
2
2
(d)  2
3
4. Evaluate the following expressions.
(a)
(c)
8 2  4 3
6
1
2
2
2
1
 ( ) 3
2
(b) (
(d)
2 2
1
5
2 3
32
)2
 ( 2  3 2 ) 2
5. Simplify the following expressions, and express your answers in positive indices.
1
(a) (a) 0
(b) ( 4a 0 ) 2
3
2
2
(c) 2
(d) ( 5 ) 3
x
x
Ex.1B Advanced Set
2. Evaluate the following expressions.
1.8
New Trend Mathematics S3A — Junior Form Supplementary Exercises
6. Simplify the following expressions, and express your answers in positive indices.
(a) a 7  a 3
(b) a 3  a 5
(c) a 8  a 2  a 4
(d) a 1  a 3  a8
7. Simplify the following expressions, and express your answers in positive indices.
(a) (
x 3
y
2
) 2
(c) ( x  2) 1
(b) (4 x 2 y 3 ) 3
(d) 4( x  3) 2
Level 2
8. Evaluate the following expressions.
010 23
(a) ( 0   4 ) 2
10
3
(b) [(1)100  (100 ) 1 ]1
(c) [(34 ) 0  6 2 (32 )]3
(d) [(2) 3  2 3 ]10
Ex.1B Advanced Set
9. Express the following algebraic fractions in the form of x m yn where m and n are integers.
1
x
(a)
(b)
y
xy
(c)
1
x 2 y 4
(d)
x 4
y 5
10. Simplify the following expressions, and express your answers in positive indices.
3
(a) (2a 1b 3 )( a 2b)
(b) (7 2 a 4 b 3 )(7 3 a 5b 2 )
4
a 4 b 7
(c) (3ab 5 )(32 a 2 b 2 )
(d) (8 2 a12b 6 )  (
)
4
11. Simplify the following expressions, and express your answers in positive in dices.
4b 2
(2a 3b) 4 1
(a) [ 6 2 ]2
(b) [
]
a ( a b)
(5a 3b 2 ) 0
18
(c) [(3a 4b 2 ) 3 ( )]3
(d) [2 3 (b 4 c 5 ) 2 b 5 c 2 ]2
ab
12. Simplify the following expressions, and express your answers in positive indices.
(a)
(c)
( p 1 ) 2 (q 1 ) 3
p 2 q 2
(3 pq) 2 (2 p 2 q 1r ) 3
(6 p 2 q) 4
(b)
(d)
( p 3 rq 2 ) 4 (4 p 3 q 2 )
5( p 2 q 2 r ) 2
(q 3 ) 2 (4 p 3 qr) 0 (5 p 2 q 3 ) 1
(5 p 2 r 2 ) 0
Chapter 1 Laws of Indices
1.9
13. Express the following algebraic fractions in the form of x m yn where m and n are integers.
(c)
y
(b)
x 2 y 1
( xy) 3
x6 y3
(d) (
xy 2 x 3
14. Given that n is an integer, simplify
( xy 2 ) 1
x2
x 3 y 2 x
) 2
23 n
.
4 5  n
Ex.1B Advanced Set
(a)
15. Arrange the following numbers in descending order.
(a) 81 20, 27 27, 9 39, 3 79
1
1
(b) 0.25 13 , 0.549 , ( ) 16 ,
8
(16)14
Exercise 1C
 Elementary Set
Level 1
1. Solve the following exponential equations.
(a) 4 2 x  1
(c)
1
7
x
1

(b) (2  3  4) x  1
1
(d) ( ) x  4 0
4
(a) 9 x  9 100
(b) 63x  612
(c) 2 x  3  27
(d) 82 x1  85
3. Solve the following exponential equations.
(a) 8 x  82  83
(b) 10 4 x  10 2 1010
(c) 9 x  36
(d) 25 x3  510
4. Solve the following exponential equations.
(a) 2 x  16
6x
 36
(c)
6
5. Solve the following exponential equations.
1
1
(a) 2 x 
216
6
1
(c) x  25
5
(b) 10 x  10 000 000
(d) 3 12 x  432
(b) 4 2 x 
(d)
1
3
x 1
1
16
 81
Ex.1C Elementary Set
2. Solve the following exponential equations.
1.10
New Trend Mathematics S3A — Junior Form Supplementary Exercises
6. Solve the following exponential equations.
(a) 2 x  23  0
(c) 3 x1  9  0
1
(b) 33 
0
3
(d) 5(4  x )  80  0
x2
Level 2
Ex.1C Elementary Set
7. Solve the following exponential equations.
(a) 32x  32 x8
(b) 5(2 x3)  54 x9
(c) 4 2 x 1  (4 3 ) 2 x 5
(d) 3612 x  63 x
8. Solve the following exponential equations.
(a) (32 x )(33 x )  310
(b) (113 x )(112 x 3 )  121
(c) (25 x )(5 x 1 )  5 2
(d)
4x
2 x1
 64
9. Solve the following exponential equations.
(a) 2 x 1  2 x  2 3
(b) 3 x  3 x1  4
(c) 25(5 x1 )  5 x1  130  0
(d) 2 x 2  2 x 1  3(2 5 )  0
 Advanced Set
Level 1
1. Solve the following exponential equations.
3
(a) ( ) x  10 0
2
(c) 7 x  6  7 3

(b) 8 4x  8 20
1
1
(d) ( ) 2 x3  ( ) 1
2
2
2. Solve the following exponential equations.
Ex.1C Advanced Set
(a) 12 5 x  12 5  12 20
(b) 5 x  4  (52 )3
(c) 642 x  46
(d) 4 x3  210
3. Solve the following exponential equations.
(a) 8 x  512
(c) 10(5 x )  250
4. Solve the following exponential equations.
1

(a) 2 x  4 
625
5
1
(c) x  49
7
(b) 6 2x  216
4
(d) (9 x1 )  108
3
(b) 32 x 
1
81
1
(d) ( ) 3 x  64
4
Chapter 1 Laws of Indices
1.11
5. Solve the following exponential equations.
(a) 10 3x  10 2  0
(b) 2 2 x1  32  0
6
(d) x  384  0
4
(c) 52 x1  125  0
Level 2
6. Solve the following exponential equations.
7. Solve the following exponential equations.
(a) 7 2 x  75 x  7 7
(c) 512 x  5 x2  125
8. Solve the following exponential equations.
92x
(a) x2  8110
3
243
(c) 9 x  x  0
27
(b) 3x2  32 x  311  0
1
(d) 6 x  63 x6 
0
36
(b) 10 2 x 1 
(d)
1
10 x 1
Ex.1C Advanced Set
(b) 52 x8  625 x4
1
 125 x 1  0
(d)
x4
25
(a) 2 93 x  2123 x
1
(c) x 3  32 x 5
9
0
42x
512
 x1  0
16 32
9. Solve the following exponential equations.
(a) 3x1  3x  18
(c) 5 x2  2  5 x1 
(b) 4 x1  4 x1 
3
0
25
15
256
(d) 18  32 x4  9 x2  171  0
Exercise 1D
 Elementary Set
Level 1

(a) 10 (
)
 1 000 000 000
(b) 10 (
)
 100 000
(c) 10 (
)
1
(d) 10 (
)
 0.01
(
)
 0.000 001
(
)
 0.000 000 1
(e) 10
(f) 10
2. Express the following numbers in scientific notation.
(a) 123 000
(b) 980 000 000 000
(c) 0.013 5
(d) 0.000 079 76
Ex.1D Elementary Set
1. Fill in the blanks with suitable numbers.
1.12
New Trend Mathematics S3A — Junior Form Supplementary Exercises
3. Express the following numbers as integers or decimal numbers.
(a) 1.2 10 3
(b) 2.48 108
(c) 3.75 10 4
(d) 1.864 10 12
4. Evaluate the following and express your answers in scientific notation. (Correct your
answers to 3 significant figures if necessary.)
(a) 2 400  0.2
(b) 0.005 4  2 000 000
(c) (1.1 10 )  (3  10 )
3
(e)
(g)
(i)
4
1.2  10 8
2  10 4
2.4  10 13
9.6  10 7
7.41  10 6
3  10 18
(d) (4.25  10 5 )  (6.08  10 2 )
(f)
(h)
(j)
4.8  10 3
8  10 4
1.23  10 6
3.69  1017
(2  10 21)  (3.88  10 25 )
1.94  10 34
5. Evaluate the following and express your answers in scientific notation. (Correct your
answers to 3 significant figures if necessary.)
Ex.1D Elementary Set
(a) 1 000 2
(b) ( 2  10 2 ) 3
(c) (3  10 5 ) 2
(d) (12  10 6 ) 2
(e) (0.01  10 4 ) 4
(f) (0.252  10 5 ) 3
(g) (2  10 8 )  (0.08  10 10 )
(h) (3.48  10 9 )  (1.65  10 21 )
(i)
0.04  10 4
0.000 2
(j)
(4.8  10 6 )  10 4
0.000 8  10 12
6. Given that the speed of light in a vacuum is 299 792 458 m/s, express it in the same unit
and in scientific notation. (Correct your answer to 4 significant figures.)
7. Given that the radius of the Earth is approximately 6 380 km, express it in m and in
scientific notation.
8. Express the values of the following as integers or decimal numbers.
(a) (1.8  10 5 )  (0.4  10 2 )
(c)
6.84  10 8
4  10 4
(b) (1.5  10 2 )  (3.6  10 4 )
(d)
7.24  10 10
0.8  10 6
Level 2
9. Evaluate the following by using a calculator, and express your answers in scientific
notation. (Correct your answers to 3 significant figures if necessary.)
(a) 225  324
(c) 6.57  9.8112
(b) 21 200 5
5 007 6 009
(d)

3
7
1.13
10. Express the following numbers in scientific notation, and arrange them in descending order.
(Correct your answers to 3 significant figures if necessary.)
65 , 56 , 47 , 38 , 29
11. The water consumption in a city was 57 284 300 000 m 3. If there were 365 days last year,
find the average daily water consumption in scientific notation. (Correct your answer to
3 significant figures.)
 Advanced Set
Level 1
Ex.1D Elementary Set
Chapter 1 Laws of Indices

1. Express the following numbers in scientific notation.
(a) 0.001 49
(b) 4 729 000 000
(c) 0.000 000 94
(d) 103 400 000
2. Evaluate the following and express your answers in scientific notation. (Correct your
answers to 3 significant figures if necessary.)
(a) 52 000  0.03
(b) 0.000 57  6 410 005
(e)
(g)
(i)
3.8  1014
2  10 2
2.4  10 10
3.6  10 9
3.81  10 9
2.46  10 21
(d) (3.56  10 7 )  (2  10 3 )
(f)
(h)
(j)
5.5  10 8
1.1  10 20
1.5  1015
7.5  10 36
(2  10 5 )  (1.66  1010 )
2.84  10 30
3. Evaluate the following and express your answers in scientific notation. (Correct your
answers to 3 significant figures if necessary.)
(a) 4 000 2
(b) 200 3
(c) (2  10 4 ) 4
(d) (0.25  10 4 ) 4
(e) (16  10 8 )  (0.25  10 20 )
(f) (2.5  10 2 )  (0.2  10 7 )
(g) (360  10 7 )  (1.45  10 5 )
(h)
(i)
13.2  10 8
11  10 2
(j)
0.03  10 4
0.000 9
(96  10 8 )  (0.48  10 5 )
(0.000 16  1010 )  10 3
4. Given that there were approximately 6 079 600 000 people living on Earth in year 2000,
express it in scientific notation. (Correct your answer to 4 significant figures.)
Ex.1D Advanced Set
(c) (1.8  10 3 )  (2  10 5 )
1.14
New Trend Mathematics S3A — Junior Form Supplementary Exercises
5. Given that the length of a river is 6 211.3 km, express it in cm and in scientific notation.
6. Express the values of the following as integers or decimal numbers.
(a) 1.31 10 6
(c) (9.4  10 3 )  (5  1010 )
(b) 3.65 10 4
0.86  1012
(d)
0.4  1015
Level 2
7. Evaluate the following by using a calculator, and express your answers in scientific
notation. (Correct your answers to 3 significant figures.)
(a) 310  210
2 004 2 005
(c)

5
6
1 997
(e)
1.99  10 7
(b) 91 2 005 2
(d) (1.97  10 7  2) 2
(f) (
46 5  58 7
46
5
 58
7
) 2
Ex.1D Advanced Set
8. Express the following numbers in scientific notation, and arrange them in ascending
order. (Correct your answers to 3 significant figures.)
22 22 , 2321, 24 20 , 2519
9. Express the following numbers in scientific notation, and arrange them in descending order.
(Correct your answers to 3 significant figures if necessary.)
139 , 1211, 1110 , 109 , 98
10. Great Land Development Company plans to demolish seven buildings in Honey Garden. It
will produce about 200 thousand tonnes (1 tonne = 1 000 kg) of demolition waste and cost
about $2.5 million to deal with it. Express the cost to deal with the demolition waste per kg
in scientific notation.
11. It is known that the distance for light travelling in a vacuum in one year (365 days) is one
light year.
(a) (i) How many seconds are there in a year?
(ii) If the speed of light in a vacuum is 3.00  10 8 m/s, express the distance of one light
year in km and in scientific notation. (Correct your answer to 3 significant figures.)
(b) Given that the shortest distance between two particular planets is 16 light years,
express it in km and in scientific notation. (Correct your answer to 3 significant
figures.)
(c) A railway express travels 135 km/h. If we travelled between the two planets in (b) at
this speed, how long would it take for a single journey? (Express your answer in
scientific notation.)
Chapter 1 Laws of Indices
Exercise 1E
 Elementary Set
Level 1
1.15

1. Fill in the blanks with suitable numbers.
(a) (
)  3 000  200  70  6
(b) (
)  20 000  4 000  10  7
(c) 9 876  9 000  800  (
)6
(d) 80 686  80 000  (
)  80  6
(e) 14 641  10 000  4 000  (
)  40  1
 1  10  4  10  (
4
)  10(
3
)
 4  10 1  1  10 0
2. Fill in the blanks with suitable numbers.
(a) (
)  40  1  0.9  0.05
(b) (
)  20  3  0.4  0.006
)  0.005
(d) 73.072  70  3  (
)  0.002
(e) 53.782  50  3  0.7  (
)  0.002
1
0
1
 5  10  3  10  7  10  (
)  10 (
)
 2  10 3
3. Write down the place value of each digit of 20 469.
Digit
2
0
4
Place value
6
9
100
4. Write down the place value of each digit in 16 384 in the index form with base 10.
Digit
1
Place value
6
10
3
8
4
9
2
3
5. Write down the place value of each digit in 0.379 2.
Digit
0
3
Place value
7
0.01
6. Write down the place value of each digit in 80.297 in the index form with base 10.
Digit
Place value
8
0
2
10
1
9
7
Ex.1E Elementary Set
(c) 12.345  10  2  0.3  (
1.16
New Trend Mathematics S3A — Junior Form Supplementary Exercises
7. Write down the place value of each digit 3 in the following denary numbers.
(a) 5 632
(b) 23 420
(c) 702.32
(d) 304.523
8. Express the following numbers in the expanded form with base 10.
(a) 12
(c) 37 441
(b) 323
(d) 20 500
9. Express the following as denary numbers.
Ex.1E Elementary Set
(a) 1  10  5  1
(b) 3  100  5  10  1  1
(c) 7  10 000  3  1 000  5  1
(d) 5  10  6  1  2  0.1  3  0.01
Level 2
10. Use all of the following numerals (without repetition) to form the greatest denary numbers.
(a) 3, 2, 9, 7
(c) 1, 0, 9, 2, 9
(b) 6, 3, 1, 0, 8
11. Use all of the following numerals (without repetition) and a decimal point to form the
smallest denary numbers.
(a) 1, 8, 4
(b) 1, 5, 6, 0
(c) 0, 8, 6, 9, 0
12. In each of the following, how many times is the place value of the left -most digit 4 to that
of the right-most digit 4?
(a) 4 004
(b) 4 040
(c) 34.004
 Advanced Set
Level 1

1. Fill in the blanks with suitable numbers.
Ex.1E Advanced Set
(a) (
)  90 000  4 000  300  80  1
(b) (
)  80 000  300  90  5
(c) 45 091  40 000  5 000  (
)1
(d) 27 504  20 000  7 000  (
 2  10 4  (
)4
)(
)(
)
Chapter 1 Laws of Indices
1.17
2. Fill in the blanks with suitable numbers.
(a) (
)  10  3  0.8  0.04
(b) (
)  50  9  0.5  0.003
(c) 63.054  60  3  (
)  0.004
(d) 49.827  40  9  0.8  (
 4  10 1  9  100  (
)(
)
)(
)(
)
3. Write down the place value of each digit in 96 573.
Digit
9
6
5
7
Place value
3
10
4. Write down the place value of each digit in 1.720 4 in the index form with base 10.
1
Place value
10
7
2
0
4
0
5. Write down the place value of each digit 4 in the following denary numbers.
(a) 4 342
(b) 43 423
(c) 74 421
(d) 4 001.342
6. Express the following numbers in the expanded form with base 10.
(a) 25
(b) 111
(c) 10 792
(d) 31.415
7. Express the following as denary numbers.
(a) 1  100  3  10  7  1
(b) 7  1 000  4  100  8  1
(c) 8  100 000  5  10 000  4  100
(d) 2  10 1  7  100  1  10 1  4  10 2  5  10 3
Level 2
8. Use all of the following numerals (without repetition) to form (i) the greatest denary
numbers; and (ii) the smallest denary numbers.
(a) 3, 9, 8, 4
(c) 1, 7, 9, 6, 2
(b) 7, 1, 5, 0
(d) 5, 0, 6, 5, 2
9. Use the numerals 0, 1, 2, 3 and 4 (without repetition) to form (a) the greatest denary odd
number; and (b) the smallest denary odd number.
Ex.1E Advanced Set
Digit
1.18
New Trend Mathematics S3A — Junior Form Supplementary Exercises
10. Use all of the following numerals (without repetition) and a decimal point to form the
smallest denary numbers.
(a) 6, 5, 8, 3
(b) 2, 4, 0, 9, 1, 5
(c) 7, 7, 1, 8
(d) 5, 0, 9, 7, 0, 0
Ex.1E Advanced Set
11. In each of the following numbers, how many times is the place value of the left -most
digit 6 to that of the right-most digit 6?
(a) 6 036
(b) 460 246
(c) 46 087 678
(d) 8.636
(e) 96.086 4
(f) 1.960 864
12. It is given six numerals 0, 3, 1, 2, 5 and 8.
(a) (i) Use all numerals above (without repetition) to form (I) the greatest denary number;
and (II) the smallest denary number.
(ii) Write down the respective place values of digit 8 in the two numbers in (a)(i).
(b) (i) Use all numerals above (without repetition) and a decimal point to form the
smallest denary number.
(ii) Write down the place value of digit 8 in the number in (b)(i).
Exercise 1F
 Elementary Set
Level 1

1. Find the values of the following numbers.
(a) 2 0
(b) 2 5
(c) 2 8
(d) 2 10
Ex.1F Elementary Set
2. Fill in the blanks by using either 0 or 1 to make the following expressions correct.
(a) 9  1  8  0  4  (
(b) 13  1  8  (
)2(
)4(
)1
)211
(c) 37  1  32  (
)  16  (
)8(
)4(
(d) 102  (
)  64  (
)  32  (
)  16  (
(
)2(
)1
)2(
)1
)  8 (
)4
3. Fill in the blanks with suitable numbers.
(a) 1 2  2(
(c) 100 2  2(
(b) 10 2  2(
)
(d) 1 000 2  2(
)
(e) 100 000 2  2(
)
)
)
(f) 10 000 000 2  2(
)
Chapter 1 Laws of Indices
1.19
4. Write down the place value of each digit in 10 101 2 in the index form with base 2.
Digit
1
0
1
0
1
Place value
5. Write down the place value of each digit in 1 101 2.
Digit
1
1
0
1
Place value
6. Write down the place value of each digit 0 in the following binary numbers.
(a) 1 011 2
(b) 1 011 1112
(c) 1 110 1112
(d) 101 101 2
7. Fill in the blanks with suitable numbers to make the following expressions correct.
(b) 110 2  (
(c) 11 010 2  (
)  20
)  2 2  1  2(
)  2 4  1  2(
)
 0  20
)
 0  2(
)
(
)  21  (
8. Express the following binary numbers in the expanded form with base 2.
(a) 11 2
(b) 1 011 2
(c) 1 001 2
(d) 10 111 2
9. Express the following as binary numbers.
(a)
(b)
(c)
(d)
1  22  1  21  1  20
1  23  0  22  0  21  1  20
1  25  0  24  1  23  0  22  0  2  0  1
1  26  1  25  1
10. Convert the following binary numbers into denary numbers.
(a) 111 2
(b) 1 110 2
(c) 11 010 2
(d) 101 100 2
(e) 1 001 100 2
(f) 1 111 000 2
11. Convert the following denary numbers into binary numbers.
(a) 9
(b) 36
(c) 86
(d) 101
(e) 215
(f) 543
)  20
Ex.1F Elementary Set
(a) 100 2  1  2 2  0  2 1  (
1.20
New Trend Mathematics S3A — Junior Form Supplementary Exercises
Level 2
Ex.1F Elementary Set
12. In each of the following binary numbers, how many times is the pla ce value of the
left-most digit 0 to that of the right-most digit 0?
(a) 1 010 2
(b) 100 111 2
(c) 11 000 2
(d) 1 000 101 2
13. Write down the greatest and the smallest 4-digit binary numbers which has only one 0 in
the respective numbers.
 Advanced Set
Level 1

1. Find the values of the following numbers.
(a) 2 2
(b) 2 7
(c) 2 9
(d) 2 12
2. Fill in the blanks with suitable numbers.
(a) 10 000 2  2(
(b) 1 000 000 2  2(
)
(c) 100 000 000 2  2(
)
(d) 100 000 000 000 2  2(
)
)
3. Write down the place value of each digit in 110 011 2 in the index form with base 2.
Ex.1F Advanced Set
Digit
1
1
0
0
1
1
Place value
4. Write down the place value of each digit in 10 011 2.
Digit
1
0
0
1
1
Place value
5. Write down the place value of each digit 1 in the following binary numbers.
(a) 1 001 2
(b) 100 100 2
(c) 100 010 2
(d) 11 000 2
6. Fill in the blanks with suitable numbers to make the following expressions correct.
(a) 111 2  1  2 2  (
(b) 1 100 2  1  2(
(c) 101 001 2  1  2(
)  21  1  2 0
)
(
)
(
)  22  (
)(
)  2 1  0  2(
)(
)(
)
)  1  20
Chapter 1 Laws of Indices
1.21
7. Express the following binary numbers in the expanded form with base 2.
(a) 101 2
(b) 1 010 2
(c) 10 100 2
(d) 11 001 001 2
8. Express the following as binary numbers.
(a) 1  2 2  0  2 1  0  2 0
(b) 1  2 6  1  2 4  1  2 2  1
(c) 1  16  1  8  0  4  0  2  1
(d) 1  128  1  64  1  32  1  2
9. Convert the following binary numbers into denary numbers.
(a) 1 010 2
(b) 11 101 2
(c) 100 101 2
(d) 1 100 100 2
(e) 11 100 111 2
(f) 111 001 010 101 2
10. Convert the following denary numbers into binary numbers.
(b) 110
(d) 693
Level 2
11. In each of the following binary numbers, how many times is the place value of the
left-most digit 0 to that of the right-most digit 0?
(a) 1 001 2
(b) 110 011 2
(c) 1 011 101 2
(d) 10 100 101 2
12. Write down the greatest and the smallest 6-digit binary numbers which has only three 0’s in
the respective numbers.
13. (a) Convert binary numbers 1 100 2, 1 1102, 101 0102 and 111 111 2 into denary numbers.
(b) Hence express the value of the following expression as binary number.
1 100 2  1 110 2  101 0102  111 111 2
14. The concept of decimal numbers also exists in the binary system. The following table
shows the place value of each digit in 10.101 12 .
Digit
1
Place value
2
1
0
2
0
1
2
1
0
2
2
1
2
3
1
2
4
Hence 10.101 1 2 can be converted into denary number by the following method.
10.101 12  1  2 1  0  2 0  1  2 1  0  2 2  1  2 3  1  2 4
 2  0  0.5  0  0.125  0.062 5
 2.687 5
Ex.1F Advanced Set
(a) 27
(c) 479
1.22
New Trend Mathematics S3A — Junior Form Supplementary Exercises
Ex.1F Advanced Set
(a) Convert the following binary numbers into denary numbers.
(i) 0.01 2
(ii) 11.1 2
(iii) 101.011 2
(b) Convert 0.75 into binary number.
[ Hint: 0.75  0.5  0.25  2 1  2 2 ]
(c) Convert 7.562 5 into binary number.
Exercise 1G
 Elementary Set
Level 1

1. Complete the following table.
Basic numerals in hexadecimal system
0
3
8
A
C
F
Corresponding values in denary system
2. Fill in the blanks with suitable numbers to make the following expressions correct.
(a) 18  1  16  (
(b) 80  (
)1
)  16  (
)1
(c) 305  1  256  (
(d) 1 285  (
)  16  (
)  256  (
)1
)  16  (
)1
Ex.1G Elementary Set
3. Write down the place value of each digit in 28 A30 16 in the index form with base 16.
Digit
2
8
A
3
0
Place value
4. Write down the place value of each digit 0 in the following hexadecimals numbers.
(a) A0 16
(b) 304 16
(c) 40 A53 16
(d) 7B 0A0 16
5. Fill in the blanks with suitable numbers to make the following expressions correct.
(a) 18 16  (
)  16 1  (
(b) 2A3 16  (
)  16 2  (
(c) D 702 16  13  16(
(d) 1 BEF 16  1  16(
(e) A 5EF 16  (
(f) 4D 30B 16  (
(
)  16 0
)
)
)  16 1  (
 7  16(
(
 0  16(
)  16 (
)  16 3  5  16 (
)  16 (
)  16 (
)
)
)  16 0
)
)
(
(
)
 13  16(
(
)
(
)  16 0
)
 2  16 0
)  16 1  (
)  16 (
)
)  16 0
(
)  16 (
)  16 (
)
)
Chapter 1 Laws of Indices
1.23
6. Express the following hexadecimal numbers in the expanded form with base 16.
(a) 20 16
(b) C4 16
(c) BD3 16
(d) 6 EC7 16
(e) A 061 16
(f) B1 129 16
(c) 2A7 16
(d) AC0 16
(e) 1 7A2 16
(f) A BCD 16
Ex.1G Elementary Set
7. Convert the following hexadecimal numbers into denary numbers.
(a) 83 16
(b) ED 16
8. Convert the following denary numbers into hexadecimal numbers.
(a) 4
(b) 17
(c) 48
(d) 127
(e) 200
(f) 7 430
Level 2
9. In each of the following hexadecimal numbers, how many times is the place value of the
left-most digit A to that of the right-most digit A?
(a) AA 16
(b) A4A 16
(c) 9A 25A 16
(d) A05 7A3 16
10. Convert the following binary numbers into hexadecimal numbers.
(a) 1 010 2
(b) 1 110 2
(c) 11 011 101 2
(d) 11 111 111 2

1. Write down the place value of each digit in 2A1 B00 16 in the index form with base 16.
Digit
2
A
1
B
0
0
Place value
2. Write down the place value of each digit A in the following hexadecimal numbers.
(a) 2A 16
(b) 1 0A2 16
(c) 2A1 201 16
(d) 50A 111 16
Ex.1G Advanced Set
 Advanced Set
Level 1
1.24
New Trend Mathematics S3A — Junior Form Supplementary Exercises
3. Fill in the blanks with suitable numbers.
(a) 3 618 16  3  16 3  6  162  (
(b) D3 945 16  13  16(
(c) 5 A0F 16  5  16(
(
(d) BE 676 16  (
(
)
)
 3  16(
(
)  161  8  16 0
)
 9  16 (
)  16 2  (
)
 4  16 (
)
 5  16(
)  16 1  15  16 (
)
)
) 10
)  16 4  (
) 10
)(
)(
)  6  16 0
4. Express the following hexadecimal numbers in the expanded form with base 16.
(a) 27 16
(b) 12B 16
(c) C02 16
(d) A 1E4 16
(e) FB 13A 16
(f) CA4 32F 16
5. Convert the following hexadecimal numbers into denary numbers.
Ex.1G Advanced Set
(a) 1A 16
(b) 52F 16
(c) B11 16
(d) 2 CC9 16
(e) A B0F 16
(f) 1A BFF 16
6. Convert the following denary numbers into hexadecimal numbers.
(a) 9
(b) 24
(c) 32
(d) 469
(e) 1 600
(f) 88 999
Level 2
7. Convert the following hexadecimal numbers into denary numbers, and arrange them in
descending order.
3 DD4 16, ABF 16, F 100 16, 8 9B5 16
8. In each of the following hexadecimal numbers, how many times is the place value of the
left-most digit F to that of the right-most digit F?
(a) FFC 16
(b) 2F 34F 16
(c) 98F 07E F23 16
(d) 1F3 FFD 294 16
9. Convert the following binary numbers into hexadecimal numbers.
(a) 1 001 2
(b) 10 111 2
(c) 11 100 011 2
(d) 110 101 111 2
1.25
Chapter 1 Laws of Indices
10. Cobeian, living on planet Cobe, is an organism of high intelligence. Since they have four
hands with four fingers each, they express numbers naturally by using hexadecimal system.
The following shows the basic numerals used on planet Cobe and the corresponding basic
numerals used on Earth.
Cobe
0
!
@
#
$
^
&
*
(
)
~
<
>
?
;
:
Earth
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
(a) Convert the following numbers used on planet Cobe into denary numbers used on
Earth.
(i) # 0
(ii) @ ? $
(iii) ^ 0 ; 0
(iv) > @ <
(b) Convert the following denary numbers used on Earth into the numbers used on planet
Cobe.
(i) 97
(ii) 672
(iii) 2 560
(iv) 8 904
Also the use of a mathematic symbol ‘※’ on planet Cobe is similar to that of ‘%’ on Earth.
1
1
‘%’ on Earth is ‘
’ while ‘※’ on planet Cobe represents ‘
’ on Earth. For example,
256
100
for the expression ‘@0※’ on planet Cobe, ‘@0’ represents the number ‘20 16  32’ on Earth.
32 1
Thus ‘@0※’ represents the number ‘
 ’ on Earth.
256 8
(c) Use percentages used on Earth to express the following expressions on planet Cobe.
(i) !※
(ii) ! 0 0※
(iii) & 0※
(iv) $ ( ~※
CHAPTER TEST
(Time allowed: 1 hour)
Section A (1) [ 3 marks each ]
1. Evaluate
2. Simplify
(6 2  4 2 ) 3
(3  2) 2
12 x 5
( 16 x 2 )  4
.
, and express your answer in positive indices.
3. Solve the exponential equation 3 x1  9 x2  0 .
4. Evaluate 0.000 020 312  2 3 and express your answer in scientific notation.
Ex.1G Advanced Set
For example, ‘# ! <’ represents the hexadecimal number ‘31B 16’ on Earth, ‘(0)?’ represents
the hexadecimal number ‘8 09D 16’ on Earth.
1.26
New Trend Mathematics S3A — Junior Form Supplementary Exercises
5. Convert 1 001 110 2 into a denary number.
6. Convert 986 into a hexadecimal number.
Section A (2) [ 6 marks each ]
3 2
2 3
7. Simplify (6 x y) (4 xy ) , and express your answer in positive indices.
( 2 x 3 y 2 ) 2
8. Solve the exponential equation 5 x3  5 x1 
9. Evaluate
24
.
5
(28  103 ) 2 (7  105 ) 1
, and express your answer in scientific notation.
(20  10  2 ) 4
10. Given that n is an integer, simplify
12  32n1  2  9 n
9 n1  4  32n
.
Section B
11. The following table shows the expenditure on defence of a country from 2000 to 2004.
Year
Expenditure on defence ($1 billion)
2004
207.1
2003
185.3
2002
166.2
2001
141.0
2000
120.5
(a) (i) What is the percentage increase in the expenditure on defence of the country in this
five years?
(ii) If the expenditure on defence of the country increased by 14.3% in 2005 in
comparison with that in 2004, express the expenditure on defence of the country in
2005 in scientific notation.
(4 marks)
(b) Express the total expenditure on defence of the country from 2000 to 2005 in scientific
notation.
(4 marks)
(c) If the monthly living costs of the poor in the country is $40 per head, how many poor
people can be supported for a year by the total expenditure on defence of the country
from 2000 to 2005? Express your answer in scientific notation.
(5 marks)
(Correct your answers to 4 significant figures if necessary.)
Chapter 1 Laws of Indices
1.27
Multiple Choice Questions [ 3 marks each ]
12. Given that a is a non-zero constant, m
and n are integers, which of the
following must be correct?
16. Solve the exponential equation 6 x  36 2 .
A. 6
a m  a n  a mn
II. (a m ) n  a m n
B. 4
III. (a  b) m  a m  b m
1
IV. ( ) m  a m
a
A. I and III only
D. 2
I.
C. 3
17. Solve the exponential equation
7(2 x )  2 x 1  26 .
B. I and IV only
A. 2
C. II and IV only
B. 4
26
C.
7
26
D.
6
D. I, III and IV only
□
13. Which of the following have the same
value?
I. 0.25 30
II. 2.530
III. 4
8.2  10 7
7
II.
 10 3
1 000
III. 0.34  10 5
I.
A. I and II only
B. I and III only
D. None of the above
□
18. Which of the following is not expressed
in scientific notation?
30
C. II and III only
□
□
A. II only
B. I and II only
14. (5a )
1

5
.
a
a
B.
.
5
C. 5a .
1
D.
.
5a
C. I and III only
D. II and III only
A.
□
□
19. Given a  1.03  108, b  0.000 07  1011,
c  123  105 and d  98 700 000 000  103,
which of the following is correct?
A. b < a < c < d
B. d < c < a < b
15. (4 1 ab 2 ) 2 (2 2 a 2 b 3 ) 2 
C. b < c < d < a
D. a < c < d < b
A. 1.
B. a 3b .
C. a 6b 2 .
D. 256 a 2 b10 .
□
□
1.28
New Trend Mathematics S3A — Junior Form Supplementary Exercises
20. Given a  6 201, b  36 99 and c  216 68,
which of the following is correct?
24. Given a  1 000 0002, b  4116 and c  6010,
which of the following is correct?
A. a < b < c
A. c > a > b
B. c < b < a
B. a > b > c
C. b < a < c
C. a > c > b
□
D. b < c < a
21. Given that
2 n2  2 n1
A.
B.
C.
D.
2 n1
7
.
2
5
.
4
5
 .
4
7
 .
2
n
is
an
integer,
then

D. b > a > c
□
25. Which of the following has the smallest
value?
A. (3.2  102) 2
B. 1 000 000 000 2
C. 256 16
D. 640 10
□
□
26. Which of the following has the value
equal to that of A07 16?
I. 1 007 10
II. 2 567 10
III. 100 000 1112
22. Which of the following must not be a
binary number?
I. 11 111
II. 10 100
III. 12 012
B. II only
D. I and III only
□
23. What is the place value of digit F in
1 2FB 16?
A. 10
B. 15
C. 16
D. 256
A. I and III only
B. I and IV only
C. II and III only
D. II and IV only
A. I only
C. III only
IV. 101 000 000 111 2
□
□