# 1.Evaluate the Following Expressions Chapter 1Laws of Indices 1.1

Chapter 1Laws of Indices

Warm-up Exercise

1.Evaluate the following expressions.

(a)634232(b)122(2343)

2.Evaluate the following expressions.

(a)(b)

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33(b)(5)656

6.Express the following expressions in index notation.

(a)2533(b)xy(x)z(z)(y)x

Build-up Exercise

[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Setand Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 1A

Elementary Set

Level 1

1.Simplify the following expressions.

(a)(b)

(c)(d)

2.Simplify the following expressions.

(a)(b)

(c)(d)

3.Simplify the following expressions.

(a)(b)

(c)(d)

4.Simplify the following expressions.

(a)(b)

(c)(d)

5.Given that nis a positive integer, simplify the following expressions.

(a)(b)

(c)(d)

Level 2

6.Simplify the following expressions.

(a)(b)(c)

7.Simplify the following expressions.

(a)(b)(c)

8.Simplify the following expressions.

(a)(b)(c)

9.Simplify the following expressions.

(a)(b)(c)

10.Simplify the following expressions.

(a)(b)(c)

11.Given that n is a positive integer, simplify the following expressions.

(a)(b)(c)

12.Given that n is a positive integer, simplify the following expressions.

(a)(b)(c)

13.Given that m is a positive integer wherem1, simplify.

Level 1

1.Simplify the following expressions.

(a)(b)

(c)(d)

2.Simplify the following expressions.

(a)(b)

(c)(d)

3.Simplify the following expressions.

(a)(b)

(c)(d)

4.Simplify the following expressions.

(a)(b)

(c)(d)

5.Given that n is a positive integer where, simplify the following expressions.

(a)(b)

(c)(d)

(e)(f)

Level 2

6.Simplify the following expressions.

(a)(b)

(c)(d)

7.Simplify the following expressions.

(a)(b)

(c)(d)

8.Simplify the following expressions.

(a)(b)

(c)(d)

9.Simplify the following expressions.

(a)(b)

(c)(d)

10.Simplify the following expressions.

(a)(b)

(c)(d)

11.Simplify the following expressions.

(a)(b)

(c)(d)

12.Given that n is a positive integer where, simplify the following expressions.

(a)(b)

13.Given that n is a positive integer, simplify the following expressions.

(a)(b)

(c)(d)

14.Given that n is a positive integer where, simplify the following expressions.

(a)(b)

(c)(d)

15.Arrange the following numbers in descending order.

(a)4100, 868, 1649, 3241(b),,,

Exercise 1B

Elementary Set

Level 1

1.Evaluate the following expressions.

(a)(b)

(c)(d)

2.Evaluate the following expressions.

(a)(b)

(c)(d)

3.Evaluate the following expressions.

(a)(b)

(c)(d)

4.Evaluate the following expressions.

(a)(b)

(c)(d)

5.Evaluate the following expressions.

(a)(b)

(c)(d)

6.Evaluate the following expressions.

(a)(b)

(c)(d)

7.Simplify the following expressions.

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

Level 2

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

14.Which number, 2516, 533 and 12512, is the greatest?

Level 1

1.Evaluate the following expressions.

(a)(b)

(c)(d)

2.Evaluate the following expressions.

(a)24(b)(3)3

(c)53(d)(4)2

3.Evaluate the following expressions.

(a)0.41(b)

(c)(d)

4.Evaluate the following expressions.

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

Level 2

8.Evaluate the following expressions.

(a)(b)

(c)(d)

9.Express the following algebraic fractions in the form of xmyn where m and n are integers.

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(a)(b)

(c)(d)

13.Express the following algebraic fractions in the form of xmyn where m and n are integers.

(a)(b)

(c)(d)

14.Given that n is an integer, simplify.

15.Arrange the following numbers in descending order.

(a)8120, 2727, 939, 379(b),,,

Exercise 1C

Elementary Set

Level 1

1.Solve the following exponential equations.

(a)(b)

(c)(d)

2.Solve the following exponential equations.

(a)9x9100(b)

(c)(d)

3.Solve the following exponential equations.

(a)(b)

(c)(d)

4.Solve the following exponential equations.

(a)(b)

(c)(d)

5.Solve the following exponential equations.

(a)(b)

(c)(d)

6.Solve the following exponential equations.

(a)(b)

(c)(d)

Level 2

7.Solve the following exponential equations.

(a)(b)

(c)(d)

8.Solve the following exponential equations.

(a)(b)

(c)(d)

9.Solve the following exponential equations.

(a)(b)

(c)(d)

Level 1

1.Solve the following exponential equations.

(a)(b)84x820

(c)7x673(d)

2.Solve the following exponential equations.

(a)(b)

(c)(d)

3.Solve the following exponential equations.

(a)8x512(b)62x216

(c)10(5x)250(d)

4.Solve the following exponential equations.

(a)(b)

(c)(d)

5.Solve the following exponential equations.

(a)(b)

(c)(d)

Level 2

6.Solve the following exponential equations.

(a)(b)

(c)(d)

7.Solve the following exponential equations.

(a)(b)

(c)(d)

8.Solve the following exponential equations.

(a)(b)

(c)(d)

9.Solve the following exponential equations.

(a)(b)

(c)(d)

Exercise 1D

Elementary Set

Level 1

1.Fill in the blanks with suitable numbers.

(a)10()1 000 000 000(b)10()100 000

(c)10()1(d)10()0.01

(e)10()0.000 001(f)10()0.000 000 1

2.Express the following numbers in scientific notation.

(a)123 000(b)980000000 000

(c)0.013 5(d)0.000 079 76

3.Express the following numbers as integers or decimal numbers.

(a)(b)

(c)(d)

(a)(b)

(c)(d)

(e)(f)

(g)(h)

(i)(j)

(a)(b)

(c)(d)

(e)(f)

(g)(h)

(i)(j)

6.Given that the speed of light in a vacuum is 299792458m/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 6380km, express it in m and in scientific notation.

8.Express the values of the following as integers or decimal numbers.

(a)(b)

(c)(d)

Level 2

(a)(b)

(c)(d)

10.Express the following numbers in scientific notation, and arrange them in descending order. (Correct your answers to 3 significant figures if necessary.)

11.The water consumption in a city was 57284300000m3. If there were 365 days last year, find the average daily water consumption in scientific notation. (Correct your answer to 3significant figures.)

Level 1

1.Express the following numbers in scientific notation.

(a)0.00149(b)4729000000

(c)0.00000094(d)103400000

(a)(b)

(c)(d)

(e)(f)

(g)(h)

(i)(j)

(a)(b)

(c)(d)

(e)(f)

(g)(h)

(i)(j)

4.Given that there were approximately 6079600000 people living on Earthin year 2000, express it in scientific notation. (Correct your answer to4 significant figures.)

5.Given that the length of a river is 6211.3km, express it in cm and in scientific notation.

6.Express the values of the following as integers or decimal numbers.

(a)(b)

(c)(d)

Level 2

(a)(b)

(c)(d)

(e)(f)

8.Express the following numbers in scientific notation, and arrange them in ascending order.(Correct your answers to 3 significant figures.)

9.Express the following numbers in scientific notation, and arrange them in descending order. (Correct your answers to 3 significant figures if necessary.)

10.Great Land Development Companyplans to demolish seven buildings in HoneyGarden. It will produce about200 thousand tonnes (1 tonne = 1000kg) of demolition waste and cost about \$2.5million 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 (365days) 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108m/s, express the distance of one light year inkm and in scientific notation. (Correct your answer to3 significant figures.)

(b)Given that the shortest distance between two particular planets is16 light years, express it in km and in scientific notation. (Correct your answer to 3 significant figures.)

(c)A railway express travels 135km/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.)

Exercise 1E

Elementary Set

Level 1

1.Fill in the blanks with suitable numbers.

(a)()3000200706

(b)()200004000107

(c)98769000800()6

(d)8068680000()806

(e)14641100004000()401

11044103()10()41011100

2.Fill in the blanks with suitable numbers.

(a)()4010.90.05

(b)()2030.40.006

(c)12.3451020.3()0.005

(d)73.072703()0.002

(e)53.7825030.7()0.002

510131007101()10()2103

3.Write down the place value of each digit of 20469.

Digit / 2 / 0 / 4 / 6 / 9
Place value / 100

4.Write down the place value of each digit in 16384 in the index form with base 10.

Digit / 1 / 6 / 3 / 8 / 4
Place value / 103

5.Write down the place value of each digit in 0.3792.

Digit / 0 / 3 / 7 / 9 / 2
Place value / 0.01

6.Write down the place value of each digit in 80.297 in the index form with base 10.

Digit / 8 / 0 / 2 / 9 / 7
Place value / 101

7.Write down the place value of each digit3 in the following denary numbers.

(a)5632(b)23420

(c)702.32(d)304.523

8.Express the following numbers in the expanded form with base 10.

(a)12(b)323

(c)37441(d)20500

9.Express the following as denary numbers.

(a)11051(b)310051011

(c)710000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(b)6, 3, 1, 0, 8

(c)1, 0, 9, 2, 9

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)4004(b)4040

(c)34.004

Level 1

1.Fill in the blanks with suitable numbers.

(a)()900004000300801

(b)()80000300905

(c)45091400005000()1

(d)27504200007000()4

2104()()()

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1019100()()()

3.Write down the place value of each digit in 96573.

Digit / 9 / 6 / 5 / 7 / 3
Place value / 10

4.Write down the place value of each digit in 1.7204 in the index form with base 10.

Digit / 1 / 7 / 2 / 0 / 4
Place value / 100

5.Write down the place value of each digit 4 in the following denary numbers.

(a)4342(b)43423

(c)74421(d)4001.342

6.Express the following numbers in the expanded form with base 10.

(a)25(b)111

(c)10792(d)31.415

7.Express the following as denary numbers.

(a)110031071

(b)71000410081

(c)81000005100004100

(d)2101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(b)7, 1, 5, 0

(c)1, 7, 9, 6, 2(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.

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

11.In each of the following numbers, how many times is the place value of the left-most digit6 to that of the right-most digit 6?

(a)6036(b)460246

(c)46087678(d)8.636

(e)96.0864(f)1.960864

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)20(b)25

(c)28(d)210

2.Fill in the blanks by using either 0 or 1 to make the following expressions correct.

(a)91804()2()1

(b)1318()4()211

(c)37132()16()8()4()2()1

(d)102()64()32()16()8()4

()2()1

3.Fill in the blanks with suitable numbers.

(a)122()(b)1022()

(c)10022()(d)100022()

(e)10000022()(f)1000000022()

4.Write down the place value of each digit in 101012 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 11012.

Digit / 1 / 1 / 0 / 1
Place value

6.Write down the place value of each digit 0 in the following binary numbers.

(a)10112(b)10111112

(c)11101112(d)1011012

7.Fill in the blanks with suitable numbers to make the following expressions correct.

(a)1002122021()20

(b)1102()2212()020

(c)110102()2412()02()()21()20

8.Express the following binary numbers in the expanded form with base 2.

(a)112(b)10112

(c)10012(d)101112

9.Express the following as binary numbers.

(a)122121120

(b)123022021120

(c)1250241230220201

(d)1261251

10.Convert the following binary numbers into denary numbers.

(a)1112(b)11102

(c)110102(d)1011002

(e)10011002(f)11110002

11.Convert the following denary numbers into binary numbers.

(a)9(b)36

(c)86(d)101

(e)215(f)543

Level 2

12.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)10102(b)1001112

(c)110002(d)10001012

13.Write down the greatest and the smallest 4-digit binary numbers which has only one 0 in the respective numbers.

Level 1

1.Find the values of the following numbers.

(a)22(b)27

(c)29(d)212

2.Fill in the blanks with suitable numbers.

(a)1000022()(b)100000022()

(c)10000000022() (d) 10000000000022()

3.Write down the place value of each digit in 1100112 in the index form with base 2.

Digit / 1 / 1 / 0 / 0 / 1 / 1
Place value

4.Write down the place value of each digit in 100112.

Digit / 1 / 0 / 0 / 1 / 1
Place value

5.Write down the place value of each digit 1 in the following binary numbers.

(a)10012(b)1001002

(c)1000102(d)110002

6.Fill in the blanks with suitable numbers to make the following expressions correct.

(a)1112122()21120

(b)1100212()()22()2102()

(c)101001212()()()()()120

7.Express the following binary numbers in the expanded form with base 2.

(a)1012(b)10102

(c)101002(d)110010012

8.Express the following as binary numbers.

(a)122021020(b)126124122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)10102(b)111012

(c)1001012(d)11001002

(e)111001112(f)1110010101012

10.Convert the following denary numbers into binary numbers.

(a)27(b)110

(c)479(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)10012(b)1100112

(c)10111012(d)101001012

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 11002, 11102, 1010102 and 1111112 into denary numbers.

(b)Hence express the value of the following expression as binary number.

110021110210101021111112

14.The concept of decimal numbers also exists in the binary system. The following table shows the place valueof each digit in 10.10112.

Digit / 1 / 0 / 1 / 0 / 1 / 1
Place value / 21 / 20 / 21 / 22 / 23 / 24

Hence 10.10112 can be converted into denary number by the following method.

10.10112121020121022123124

200.500.1250.0625

2.6875

(a)Convert the following binary numbers into denary numbers.

(i)0.012(ii)11.12(iii)101.0112

(b)Convert 0.75 into binary number.

[Hint: 0.750.50.252122]

(c)Convert 7.5625 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()1

(b)80()16()1

(c)3051256()16()1

(d)1 285()256()16()1

3.Write down the place value of each digit in 28A3016 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)A016(b)30416

(c)40A5316(d)7B0A016

5.Fill in the blanks with suitable numbers to make the following expressions correct.

(a)1816()161()160

(b)2A316()162()161()160

(c)D702161316()716()016()2160

(d)1BEF16116()()16()()161()160

(e)A5EF16()163516()()16()()16()

(f)4D30B16()16()1316()()16()

()16()()160

6.Express the following hexadecimal numbers in the expanded form with base 16.

(a)2016(b)C416

(c)BD316(d)6EC716

(e)A06116(f)B112916

7.Convert the following hexadecimal numbers into denary numbers.

(a)8316(b)ED16

(c)2A716(d)AC016

(e)17A216(f)ABCD16

8.Convert the following denary numbers into hexadecimal numbers.

(a)4(b)17

(c)48(d)127

(e)200(f)7430

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)AA16(b)A4A16

(c)9A25A16(d)A057A316

10.Convert the following binary numbers into hexadecimal numbers.

(a)10102(b)11102

(c)110111012(d)111111112

Level 1

1.Write down the place value of each digit in 2A1B0016 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)2A16(b)10A216

(c)2A120116(d)50A11116

3.Fill in the blanks with suitable numbers.

(a)36181631636162()1618160

(b)D3945161316()316()916()416()516()

(c)5A0F16516()()162()1611516()

()10

(d)BE67616()164()()()6160

()10

4.Express the following hexadecimal numbers in the expanded form with base 16.

(a)2716(b)12B16

(c)C0216(d)A1E416

(e)FB13A16(f)CA432F16

5.Convert the following hexadecimal numbers into denary numbers.

(a)1A16(b)52F16

(c)B1116(d)2CC916

(e)AB0F16(f)1ABFF16

6.Convert the following denary numbers into hexadecimal numbers.

(a)9(b)24

(c)32(d)469

(e)1600(f)88999

Level 2

7.Convert the following hexadecimal numbers into denary numbers, and arrange them in descending order.

3DD416, ABF16, F10016, 89B516

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)FFC16(b)2F34F16

(c)98F07EF2316(d)1F3FFD29416

9.Convert the following binary numbers into hexadecimal numbers.

(a)10012(b)101112

(c)111000112(d)1101011112

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

For example, ‘#!’ represents the hexadecimal number ‘31B16’ on Earth, ‘(0)?’ represents the hexadecimal number ‘809D16’ on Earth.

(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)2560(iv)8904

Also the use of a mathematic symbol ‘※’ on planet Cobe is similar to that of ‘%’ on Earth. ‘%’ on Earth is ‘’ while ‘※’ on planet Cobe represents‘’ on Earth. For example, for the expression ‘@0※’ on planet Cobe, ‘@0’ represents the number ‘201632’ on Earth. Thus ‘@0※’ represents the number ‘’ on Earth.

(c)Use percentages used on Earth to express the following expressions on planet Cobe.

(i)!※(ii)!00※

(iii)0※(iv)\$(~※

Chapter Test / (Time allowed: 1 hour)

SectionA(1) [3 marks each]

1.Evaluate.

3.Solve the exponential equation.

5.Convert 10011102 into a denary number.

6.Convert 986 into a hexadecimal number.

SectionA(2) [6 marks each]

8.Solve the exponential equation.

10.Given that n is an integer, simplify.

Section B

11.The following table shows the expenditure on defence of a country from 2000 to 2004.

Year / Expenditure on defence (\$1billion)
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)

Multiple Choice Questions [3 marks each]

Chapter 1Laws of Indices 1.1

12.Given that a is a non-zero constant, m and n are integers, which of the following must be correct?

I.

II.

III.

IV.

A.I and IIIonly

B.I and IVonly

C.II and IVonly

D.I, III and IVonly□

13. Which of the following have the same value?

I.

II.

III.

A.Iand II only

B.I and III only

C.IIand III only

D.None of the above□

14.

A..

B..

C..

D..□

15.

A.1.

B..

C..

D..□

16.Solvethe exponential equation.

A.6

B.4

C.3

D.2□

17.Solve the exponential equation

.

A.2

B.4

C.

D.□

18.Which of the following is not expressed in scientific notation?

I.

II.

III.

A.II only

B.I and II only

C.I and III only

D.II and III only□

19.Given a1.03108,b0.000071011,c123105and d98700000000103, which of the following is correct?

A.bacd

B.dcab

C.bcda

D.acdb□

20.Given a6201, b3699 and c21668, which of the following is correct?

A.abc

B.cba

C.bac

D.bca□

21.Given that nis an integer, then