Chapter 1 M2 Additional Math

8/12/2019 Chapter 1 M2 Additional Math

1/37

CHAPTER 1Powers

You have learnt about a power and have already known its meaning.

1. ...nn times

a a a a a

where a is any number and n is any positive integer,

na

is called a power of basea

and exponentn

.

We can call na as a to thenth

power.

Example

47 is a power of base 7 and exponent 4 .47 7 7 7 7

2,401

3)2.0( is a power in which 2.0 is a base and 3 is an exponential number.3( 0.2) ( 0.2) ( 0.2) ( 0.2)

0.008

2. 0 1a where a is any nonzero number.[Any nonzero number to the zero power is 1]

Example

09 1 0( 0.16) 1

The exponent is thenumber of times the

base is used as afactor.

Expressions writtenwith exponents are

called powers.

1.1 Properties of Powers


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2 CHAPTER 1 Powers

The property of product of powerLet a be any number and m , n be positive integers.

m n m n a a a

The property of quotient of power

Let a be any nonzero number and m , n be positive integers.

m n m n a a a

3. 1nn

aa

where a is any nonzero number and n is any positive number.

Example

33

155

55

1( 2)

( 2)

4. Product and quotient of exponents satisfy the Properties of Power as follows:

Example 4 3 4 35 5 5

75

Example 8 2 8 2( 3) ( 3) ( 3)

6( 3)


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3MATH 32201 Advanced MathematicsII

Example 1 Simplify38

740

3)3(

333

.

Solution

70 4 7 4

8 3 8

3

11 3

3 3 3 3

1( 3) 3 33

88

3)3(

7 4

8 3

3

3

3

5

3

3

3 53

23

2

1

3

1

9 Ans.

Example 2 Simplify )22()22( 3025 nnn where n is any positive integer.

Solution5 2

5 2 0 3

0 3

2 2(2 2 ) (2 2 )

2 2

n nn n n

n

5 2

3

7

3

7 3

4

2

1 2

2

2

2

2

n n

n

n

n

n n

n

nnn )25(25 22 n72

nnn )37(37 22 n42 Ans.


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4 CHAPTER 1 Powers

We have already known that scientific notationis used to represent a very high or very a

low value of positive number in the form of 10nA where 101 A and n is any integer,

for example

13

12,500,000,000,000 1.25 10 90.0000000037 3.7 10

Example 3 Rewrite7

71

106.9

)1044.1()108.4(

in the form of scientific notation.

Solution1 7 1 7

7 7

(4.8 10 ) (1.44 10 ) 4.8 1.44 10 10( ) ( )

9.6 10 9.6 10

Example 4 Simplify )105()102( 3016 and write the result in exponential form.

Solution 16 30 16 30(2 10 ) (5 10 ) (2 5) (10 10 )

16 30

14

15

10 10

10 10

10

72.0

6.9

44.18.4

1

1

2

2

0.72 (10 1)

1(7.2 ) 10

10

1 17.2 ( )

10 10

17.2

10

7.2 10

Ans.

Ans.


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Example 5 The area of Thailand is around 5 25.18 10 km .

Population and Social Research Institute of Mahidol University reported

that in January 1, 2004, the number of population in Thailand was 63.514

million people. Determine how many people per2

1 km are there onaverage?

Solution The area of Thailand is around 5 25.18 10 km .

The number of population is around 63.514 million people that is

610514.63 people.

So There are5

6

1018.5

10514.63

people per km2

1026.12 people per km2

123 people per km2 Ans.


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6 CHAPTER 1 Powers

Exercise 1.1

1. Simplify the following.1) 403 222 2) 232 )5()5()5(

3) 152 )5.0()2

1()5.0( 4) 20 )3()3(81

5) 34 )2(264 6) 024 )5

2()

5

2()

5

2(

7) 52 1010)001.0( 8) 13 192296

9) )104()105.2( 32 10) )108()1025.1( 74

11) 3 0 5 0a a a when a

12) nnn 222 53 where n is any positive integer

2. Simplify these following1) 130 )4(])4()64[(

2) 5034 )2(])2()2()2[(

3) 4225 3]333[

4) 53 )7(])7(49[

5) )109()106( 32

6) 9)103.6( 3

7) )108()104.2( 53

8) )109()106.3( 41


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8 CHAPTER 1 Powers

6. It takes 90,700 days for Pluto to rotate around the Sun.There are 41007.9 days in one year of Pluto. How

many days are there in 10 years on Pluto? Write theanswer in scientific notation.

7. The Earth is about 710296.9 miles from the Sun onaverage. Venus is about 710723.6 miles from the

Sun. On the average, how much further is the Earth

from the Venus?

8. Explain how we know which one is greater if we have 2 values in scientific notation.

For the expression, 3)2( xx , what is the value of x that makes

the expression undefined?

SI prefix

An SI ( International System of Units ) prefixis a prefix that can be applied

to an SI unit to form a decimal multiple or submultiple. Many SI prefixes

predate the introduction of the SI in 1960. They can be applied correctly to

many non-SI units. As part of the SI system they are officially determined by

theBureau International des Poids et Mesures.

SI defines a number of SI prefixesto be used with the units: these combine with

any unit name to give subdivisions and multiples. As an example, the prefix kilo

denotes a multiple of a thousand, so thekilometre is 1000metres,thekilogram

is 1000grams,a kilowatt is 1000watts,and so on. The prefix milli subdivides

by a thousand, so a millimetre is one-thousandth of a metre (1000 millimetres in

a metre), and a millilitre is one-thousandth of alitre.

Does is exist?
http://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/Bureau_International_des_Poids_et_Mesureshttp://en.wikipedia.org/wiki/Kilometrehttp://en.wikipedia.org/wiki/Kilometrehttp://en.wikipedia.org/wiki/Kilometrehttp://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Gramhttp://en.wikipedia.org/wiki/Watthttp://en.wikipedia.org/wiki/Litrehttp://en.wikipedia.org/wiki/Litrehttp://en.wikipedia.org/wiki/Watthttp://en.wikipedia.org/wiki/Gramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Kilometrehttp://en.wikipedia.org/wiki/Bureau_International_des_Poids_et_Mesureshttp://en.wikipedia.org/wiki/SI


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The prefixes are never combined; a millionth of a kilogram is a milligram, and

not a 'microkilogram'. The ability to apply the same prefixes to any SI unit is

one of the key strengths of the SI, since it considerably simplifies the system's

learning and use.

The following SI prefixes can be used to prefix any of the above units to

produce a multiple or submultiple of the original unit. This includes the degree

Celsius (e.g., 1.2 mC); however, to avoid confusion, prefixes are not used

with the time-related unit symbols min (minute), h (hour), d (day). They are not

recommended for use with the angle-related symbols (degree), (minute of

arc), and (second of arc), but for astronomical usage, they are sometimes used

with seconds of arc.

source : http://en.wikipedia.org


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10 CHAPTER 1 Powers

Product of Powers

Let a be any number, m and n be any positive integers.

We have m n m n a a a .

Then we will use this property where m and n are any integer.

Consider nm aa where 0a and m , n are the following:

1. If 0, 2m n 0 2m na a a a

0 2 21a a a

2a or 20a .

2. If 3, 0m n 3 0m na a a a

3 0

3

1 1a aa

3

1

a

3a or 03a .

3. If 4, 7m n 4 7m na a a a

4 7 4

7

1a a a

a

4 7a

3a or )7(4 a .

1.2 Operations with Powers


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4. If 8, 2m n 8 2m na a a a

8 2

8 2

1 1a a

a a

10

1

a

10a or )2(8 a .

From the above it can be noticed that the exponents of the products were formed by

summation of their exponents which is called the property of multiplication of powers.

Example 1 Simplify 1255 10 and write its product in the power form.

Solution 10 10 35 125 5 5

10 35

75

The answer is 75 .

Example 2 Simplify 54 3)3( and write its product in the power form.

Solution 4 5 54

1( 3) 3 3

( 3)

5

4

13

3

4 5

4 ( 5 )

9

3 3

3

3

The answer is 93 .

Let a be any nonzero number, and m and n be any integers.m n m na a a


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12 CHAPTER 1 Powers

Example 3 Simplify 35 3)81()3( and write its product in the power form.

Solution 5 3 5 3 3( 3) ( 81) 3 ( 3) ( 3) 3 3

4 6( 3) 3

23

The answer is 23 .

Example 4 Simplify 2537 33 aa and write its product in simplest form. ( 0a )

Solution 7 3 5 2 7 5 3 23 3 (3 3 ) ( )a a a a

2 53 a

2 5

3 a

or5

9

a

The answer is 523 a or 59 a .

Example 5 Carbon nanotube is an extremely thin hollow cylinder made of carbon atoms.

Nanotube, depending on its structure, can be metal or semiconductor. It is also

an extremely strong material and has good thermal conductivity. Its

characteristics have generated strong interest in nano-electronic and nano-

mechanical devices. For example, it can be used as nano-wires or as active

components in electronic devices such as the field-effect transistor. In 2001,

the Chinese researchers have developed carbon nanotube whose diameter is

71033 times smaller than that of human hair. If diameter of human hair is

610100 meter, what is the diameter of this carbon nanotube?

Solution Hair diameter is around 610100 m.

Diameter of nanotube is around 71033 time of hairs diameter.

Thus, the diameter of nanotube is around

7 6 7 433 10 100 10 33 10 10 m

1133 10 m

90.33 10 m

or 0.33 nm Ans.


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Exercise 1.2a

1. Simplify and write the products in the power form.1) 705 454 2)

24 42

3) 8133 83 4) 235 )2()2()2(

5) 4455 )3(333 6)4

2

3

2

1)5.0(

2

1

7) 3)5.0)(25.0( 8) 35 )4(16)4(

9) 14 )7(7)343(

10) 4 3 2( 3 ) ( 3 ) ( 3 )a a a where 0a

11) nn 6)2()2()8( where nis an integer.

12) nn aaa 490 where 0a and nis any integer.

2. Write the following expressions in terms of scientific notation.1) )106()104( 53 2) )102()104.2( 44

3) )104()105.2( 43 4) )105()102.1( 23

3. Write the following expressions in simplest form.1) 2 2 14 4 , 0a where a

2) 2 4 52 , 0a a where a

3) 7 2 8 13 3 , 0y y where y


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14 CHAPTER 1 Powers

4) 2 3 2( 3 )(2 )( ), 0b b b where b

5) 3 5 2( 2 )(5 )( ), 0c c c where c

6) 7 2 8 13 3 , 0a a where a

4. Find the value of a which makes the equation true.1) 15

10

12

2a 2)

2

4 10.25 1010

a

5. In 2003, the average world price of rice was $250per a ton. The Rice Export Union reported thatworld population needed about 400 million ton of

rice. Find the price of rice in baht? ( $1 for 40 baht )

6. Vega is the brightest star in theSummer Triangle, a group of stars is

easily visible in summer evenings in the

northern hemisphere. The name Vegais of Arabic origin meaning "stone

eagle". Vega is the fifth brightest star

in the night sky, and its diameter is

about three times that of the Sun.

If Vega is 236.5 trillion kilometers away from the Earth, how far from the Earth to the

is Vega in light years? (1 light year equal to 121046.9 km)

When we are not very ill, we usually self-medicate. Our body system has a process

to excrete the medicine in the processes of burning energy and eliminating waste.

If you take 100 milligrams of medicine, half of is excreted after every 6 hours. If you

took 1000 milligrams of medicine, how much it will remain in your body after 24

hours?

Try


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Quotient of Powers

Let a be any number, m and n be any positive integers.

We have m n m n a a a .

We will use this property where m and n are any integers.

Consider nm aa where 0a and m , n are the following.

1. If 0, 2m n 0 2m na a a a

00 2

2

aa a

a

2a or 20a .

2. If 3, 0m n 3 0m na a a a

33 0

0

aa a

a

3

1

a

3a or 03a .

3. If 4, 7m n 4 7m na a a a

44 7

7

aa a a

4

7

4 7

1

a

a

a a

11a or )7(4 a .


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16 CHAPTER 1 Powers

4. If 8, 2m n 8 2m na a a a

88 2

2

aa a

a

8

2

8 2

1

a

a

a a

6a or )2(8 a .

From the quotients above it is shown that the exponents of the quotients were formed by

summation of their exponents belong to the property of division of powers.

We have that1

nn

aa

where a is nonzero number and n is a positive integer.

From the property of division of powers we can show that 1nn

aa

where n is any

integer.

We have known thatm

m n

n

aa

a

where 0a , if 0m and n is any integer.

Then0

0 n

n

aa

a

na

and

0 1

n n

a

a a

.

Thus1n

na

a

.

This implies the following:

and 1nn

aa

.

Let a be any nonzero number, m and n be any integers.m n m na a a

Let a be any nonzero number and n be any integer, then we have

n

n

aa

1


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5

33

327

and write the answer in power form.

Solution5 3 5

8 2 8 2

27 3 3 3

3 3 3 3

3 ( 5 )

8 ( 2 )

3

3

2

10

2 ( 1 0 )

2 10

8

3

3

3

3

3

The answer is 83 .


23

ba

ba, where 0a and 0b , and write down the answer in

simplest form.

Solution3 2

3 ( 2) 2 ( 4)

2 4

a ba b

a b

3 2 2 4

5 2

a b

a b

The answer is 25ba .

Example 8 Simplifynn

nn

77

778

53

, where n is an integer, and write the answer in

simplest form.

Solution3 5 3 ( 5 )

8 8

7 7 7

7 7 7

n n n n

n n n n

2

7

( 2 ) ( 7 )

5

7

7

7

7

n

n

n n

n

The answer is

5

7

n

.


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18 CHAPTER 1 Powers

Example 9 If 1 molecule of water weighs 16100.3 kilograms, then how many

molecules are contained in 1 gram of water? Write the answer in scientific

notation.

Solution Since mass of 1 kilogram equals to 310 grams,

then 16100.3 kilograms of water equal to 316 10100.3 grams.

133 10 grams.

Since 13103 grams of water contain 1 molecule.

Hence, 1 gram of water contains13

1 1

3 10

molecules

131 103

molecules

130.333 10 molecules

or about 121033.3 molecules.

Ans.

Exercise 1.2b

1. Simplify and write the result in power form.

1)4

73

5

55 2)

3

28

3

33

3)32

272 503 4)

0

276

7

)7(7)7(

5)24

2

121)11(

11121

6)

73

24

)13()13(

131313

7)33

32

10010

)10(101000

8)008.0)004.0()2.0(

)2.0(23

2

9)53

24

)()(

aa

aaa where 0a

10)36

6263

54

n

nn

where n is an integer


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2. Write the following in scientific notation.1) )103()106( 17 2) )104()1084.2( 43

3) )00025.0()100000000005.0( 3 4)15

232

1021

)101.5()109.4(

3. Write the following in simplest form.

1) 12

10 44

64

2

256

2) )53()315( 37

3) )1111()1111( 1827 nnn where n is an integer

4) )56()8.15( 12

5)2

83

77

ab

ba where 0a and 0b

6)113-3

197

4bc2

32cb

a

a where 0a , 0b and 0c

4. UN approximated in July 2003 that the worldpopulation was around 6,300 million people and

there were about 40 percents of them lived

outside Asia, so how many people lived in

Asia?


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20 CHAPTER 1 Powers

5. About 70% of the Earths surface is covered by water. If the diameter of the Earth isabout 12,800 kilometers, how wide is the rest of the

surface of the Earth? Write the answer in scientific

notation. ( Hint: the earth surface = 4r2where r is the

radius of the Earth and let = 3.14 )

6. Rachaprapa Dam is in Surajthanee. Its old name isCheiw Lan Dam. It used for irrigation and

electricity generation. Its height is 94 m and its

bridge length is 761 m. When the dam is full the

area and capacity of the reservoir are around 185

km2and 5,638.8 cubic millionmeters, respectively.

What is the depth of if the dam is full?

7. Took stood away from the rock cliff. She shouted tothe cliff and reckoned time of her voice moving back

and forth. She found that it took 1 sec to hear her

voice which travels from her position and back. Given

that speed of voice is around 3102.1 km/hr, how far

is she from the cliff?

8. Btu (British Thermal Unit) is a unit of heat energy and kwh is a unit of electric energyused in electric fee.

For baking cake using an electric oven, it requires about 6 kwh. If this electric energy

was completely transferred to the heat energy, how many btu's were needed to bake

the cake? (Let 1 btu of heat energy equal to 41093.2 kwh of electric energy.)


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An exponent of a power

The base of a power, 58 , is 8 and its exponent is 5.

Since 8 = 32 , so we may denote 58 by 53 )2( , i.e. 53 )2( is a power of base 32 and

exponent 5.

Consider a meaning of following powers.

1. 42 )5( is a power of base 25 and exponent 4 .42

)5( =

2222

5555

= 22225

= 85

Hence, 42 )5( = 85 or 425 .

2. 32 ))3(( is a power of base 2)3( and exponent 3.32 ))3(( =

222 )3()3()3(

= )2()2()2()3(

= 6)3(

Hence, 32 ))3(( = 6)3( or 3)2()3( .

3.53

)5(

is a power of base3

5 and exponent -5.

53 )5( =53 )5(

1

=33333 55555

1

=333335

1

=155

1

Hence, 53 )5( = 155 or )5(35 .

1.3 Other Properties of Powers


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22 CHAPTER 1 Powers

4. 23 )7( is a power of base 37 and exponent -2.23 )7( =

23 )7(

1

=33 77

1

=)3()3(7

1

=67

1

Hence, 23 )7( = 67 or )2()3(7 .

It is shown that the exponents of the results above are in the from of the products of an

exponent of base and an exponent of the power. It follows the property of power.

Example 1 Find the product of243

)5(625 and write it in a form in which 25 is base.

Solution3 4 2 2 3 2 2625 (5 ) (25 ) (25 )

6 425 25

1025 Ans.

A power of base is a product of numbers

Since 14 7 2 , so 3 314 (7 2) .

Consider 3)27( , it is a power of base )27( and exponent 3.

Let consider the meaning of above condition.

Let a be any nonzero number, m and n be any integers.

( )m n m na a


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1. 3)52( is a power of base 52 and exponent 3.3)52( = )52()52()52(

= )555()222(

= 33 52

Hence, 3)52( = 33 52 .

2. 2)52( is a power of base 52 and exponent -2.2)52( =

2)52(

1

= )52()52(

1

=)55()22(

1

=22 52

1

= 22 52

Hence,2

)52(

=22

52

.

3. 0)52( is a power of base 52 and exponent 0.0)52( = 010

= 1 or 00 52

The results above follow the property of power.

Example 2 Write 315 in a form of powers of base which is a multiplication of prime

numbers.

Solution3 315 (3 5)

3 33 5 Ans.

Let a and b be any nonzero numbers, and n be any integer.

( )n n nab a b


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24 CHAPTER 1 Powers

A power of fraction base

Lets consider the meaning of following powers.

1. 37

2

is a power of base

7

2and exponent 3.

3

7

2

=

7

2

7

2

7

2

=777

222

= 3

3

7

2

Hence,

3

7

2

=

3

3

7

2.

2. 47

2

is a power of base

7

2and exponent -4.

4

7

2

= 4

7

2

1

=

7

2

7

2

7

2

7

2

1

=

4

4

7

2

1

=

4

4

2

71

=4

4

2

7

=4

4

7

2

Hence,

4

72

= 4

4

72

.


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3. 07

2

is a power of base

7

2and exponent 0.

0

7

2

= 1

=0

0

7

2

The results above follow the property of power.

Example 3 Write

5

7

3

in a form of power of base which is a fraction of prime

numbers.

Solution

5 5

5

3 3

7 7

Ans.

Example 4 Write

2

3252

7

3

37

in a form of a power of exponent of positive integer.

Solution

5 32 2 2

10 6

2

7 3 37 3

73

7

210 6

2

37 3

7

106 2

2

73 3

7

8 87 3

8

7 3

821 Ans.

Let a and b be any nonzero numbers, and n be any integer.n n

n

a a

b b


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26 CHAPTER 1 Powers

Example 5 Simplify

233

332

8

16

ba

bawhere a and b are nonzero numbers.

Solution

32 3 6 9

2 6 63 3

16 16

88

a b a b

a ba b

0 152 a b

152 1 b

152b . Ans.

Example 6 Find the volume of the Earth with radius

around 6,380,000 m. and write the answer in

scientific notation.

(Spherical volume 3

3

4rV where is around 3.14)

Solution The radius of the Earth is around 66,380,000 6.38 10 m

From 334 rV ,

where V denotes the volume of Earth, its unit is m3,

r denotes the radius of Earth which is around 61038.6 m.

Then we get 3

64 3.14 6.38 103

V m3

33 64 3.14 6.38 10

3

m3

184.1866 259.694 10 m3

181087.23 10 m3

3 181.08723 10 10 m3

211.09 10 m3.

Hence, the volume of the Earth is around 211009.1 m3. Ans.


27/37


Exercise 1.3

1. Write the following in simplest form.

1) 00 )5( 2) 063 )5( ba where 0a and 0b

3) 33 )7( 4) 1495 575

5) 321 )32( 6) 331 )24(

7) 2121 )3927( 8) 24213 584)2(

9) 221 44 a where 0a 10) 3231 )2( aaa where 0a

2. Simplify the following.

1)

22521

)2()4(

2)

272345

3223

3) 777 3212 4) 21223 104210

5) 161717 326 6) 555 2.02)4.0(

7) 1324225 )( bababa where 0a and 0b

8)

1

1

111

0

7

65

10

33

2

3

babawhere 0a and 0b

9) 2 3 3[(4 ) (9 ) ] (2 3 )m m m m where m is an integer.

10) 1 6 4[(7 ) (14 ) ] (128 7 )n n n where n is an integer.


28/37

28 CHAPTER 1 Powers

Find two numbers which cannot be divided by 10 and their product is equal to a million.

3. Find m , n and p which make the equations true.1) 49 35 5 7n m 2) 5 510 6 2 3 5m n p

4. The distance between the sun and the Earth is around149,000,000 km. Approximate the time it takes for the light

to travel from the sun to the Earth, if the speed of light is

around 300,000 km/s. Write the answer in scientific

notation.

The formulad

t

r

where t denotes the time in which light travel from the sun to the Earth.

d denotes the distance between the sun and the Earth.

r denotes the speed of light.

5. In 1887, Hertz observed the light phenomenon. Shortwave length had affected metal surface, electrical

particles from the metal surface were released. This

short wave length will leave energy particles, called

photons, to the metal surface. If we use green light of

wave length 550 nm, how much the energy from

photon do we get?

Let

hc

E where Edenotes energy of photon, joule,

(lambda) denotes wave length, nanometer(nm),

c denotes the speed of light which is equal to 8103 m/s,

h denotes the constant, 341063.6 .

TRY !

:Images obtained by Andy Hargreaves.


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Which one is true?

1. 2

01

0.0000000007 1

2. 0 1a where a is any number.

3. 1010 0

4. 4 45 ( 5)

5. 2 2(0.5) 2

6. 10 100100 10

7. 4 43 77 3

8. 10 22 10

2 10

10 2

9. 10 10 20n n n where nis any integer.

10.10 10 10m n m n where mand nare any integers.

11.10 10 1m n where mand nare integers such 0 nm .

12.10 10 1m n for the case of m = 0 and n = 0.

13. 10 10 20( )a b a b where a and b are arbitrary numbers.

Is it true?


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30 CHAPTER 1 Powers

Copyright 1999, Jim Loy

Here is the table of the names of large numbers. The system used in the U.S. is not logical

as that used in other countries (like Great Britain, France, and Germany). In other

countries, a billion (bi meaning two) has twice as many zeros as a million, and a trillion

(tri meaning three) has three times as many zeros as a million, etc. But the scientif ic

communi ty seems to use the American system.

Number of zeros U.S. & scientific community Other countries

3 thousand thousand

6 million million

9 billion 1000 million (1 milliard)

12 trillion billion

15 quadrillion 1000 billion

18 quintillion trillion

21 sextillion 1000 trillion

24 septillion quadrillion

27 octillion 1000 quadrillion

30 nonillion quintillion

33 decillion 1000 quintillion

36 undecillion sextillion

39 duodecillion 1000 sextillion

42 tredecillion septillion

45 quattuordecillion 1000 septillion

48 quindecillion octillion

51 sexdecillion 1000 octillion

54 septendecillion nonillion

57 octodecillion 1000 nonillion

60 novemdecillion decillion

63 vigintillion 1000 decillion

Million, Billion, Trillion...


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Example 1 What is the sum of 123.5 billion baht and 800 million baht?

Solution 123.5 billion is equal to 63 10105.123 baht

800 million is equal to 2 6 3 68 10 10 0.8 10 10 baht

Then 3 6 3 6123.5 10 10 0.8 10 10

3 6123.5 0.8 10 10 baht

3 6124.3 10 10 baht

So the answer is 3.124 billion baht.

Example 2 Three state agencies got budgets for their operations as follows.

12.5 billion baht 3.125 billion baht and 975 million baht

1) Find the total of the three state agencies in billion baht.

2) What is the difference between the highest budget and the lowest

budget? (in million baht)

Solution 1) The first state agency got a budget of 12.5 billion baht

or 63 10105.12 baht

or 62 1010125 baht

The second state agency got a budget of 3.125 billion baht

or 63 1010125.3 baht

or 62 101025.31 baht

The third state agency got a budget of 975 million baht

or 610975 baht

or 62 101075.9 baht

Hence, the total of the budgets of three state agencies is

62 1010125 62 101025.31 62 101075.9

2 6125 31.25 9.75 10 10

2 6166 10 10 baht

3 616.6 10 10 baht

16.6 billion baht Ans.


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32 CHAPTER 1 Powers

2) The highest budget is 12.5 billion baht

or 610500,12 baht

The lowest budget is 975 million baht

or 610975 baht

Hence, the difference is 612,500 10 610975

= 610975500,12

= 610525,11 baht

= 11,525 million baht. Ans.

Check Understanding

Simplify the following:

1) What is the sum of 2.5 billion baht and 420 million baht (in million baht)?

2) What is the difference of 12.2 billion baht from 1,320 million baht (in million baht)?

3) What is the sum of 10 trillion baht and 10 billion baht (in billion baht)?

4) What is the difference of 17 trillion baht from 20 billion baht (in trillion baht)?

5) How much larger is 47,800 million baht than 7.5 billion baht?


33/37


I=pxrxtI= $1 000 x 0.06 x 1I= $60

Each quarter is of a year.

Where you borrow money from a bank, you must pay an interest. Interest is a fee for

borrowing the money, it is a percentage charged on the principal for a period of a year -

usually. The amount of money borrowed is called the principal.

The interest you must pay, depends on how much money you borrow, the yearly interest

rate, and how long you borrow the money.

Interest = principle x rate x time

I = px

rx

t

Most savings accounts pay compound interest. Compound interest is computed at stated

period. For each period the interest earned is added to the account. At the end of each

period, interest is paid on the balance which includes the previous interest. Interest may

be added annually, semiannually (2 times a year), quarterly (4 times a year), monthly, or

daily.

If $1,000 is deposited into a savings account at 6% simplest interest, the interest earned in

one year is $60. The savings total at the end of

the first year would be $1,060.

Example 1 Find the savings total at the end of the first year if $1,000 is deposited in

savings account at 6% annual interest compounded quarterly.

Solution

Fir st quarter: I p r t

1$1000 0.06

4I

The new balance is1 0.06

$1000 $1000 0.06 $1000 14 4

after one

quarter.

After each of next three quarters, interest is computed and added to the account.

Compound Interest


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34 CHAPTER 1 Powers

Second quar ter :0.06 1

$1000 1 0.064 4

I

new balance:

4

106.0406.011000$

406.011000$

2

0.06 0.06$1000 1 1

4 4

0.06$1000 1

4

Thi rd quarter:

20.06 1

$1000 1 0.064 4

I

new balance:

4

106.0

4

06.011000$

4

06.011000$

22

2

3

0.06 0.06$1000 1 1

4 4

0.06$1000 1

4

Fourth quarter:

30.06 1

$1000 1 0.064 4

I

new balance:

4

106.0

4

06.011000$

4

06.011000$

33

3

4

0.06 0.06$1000 1 1

4 4

0.06$1000 1

4

The balance at the end of one year is

4

40.06$1000 1 $1000 (1.015)4

$1,061.36 . Ans.


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Below is a formula that can be used to find the total savings, S for given principal P,

annual interest rate r, and time tyears.

1

n t

rS Pn

wherenis the number of times interest is compounded in one year.

Example 2Find the total savings for 1,200 baht deposited in an account that pays 9%

annual interest compounded monthly for 3 years.

Solution

12 30.09

1200 1

12

S

36

1200 0.0075

The total savings is 1,570.37 baht. Ans.

From the formula, where nis equal to 1, we have

1 t

S P r .

Notice that

t

r

1 is a power of base r

1 and exponentt

. Since 0

r , then 11 r .

If we let tbe a horizontal axis and S be a vertical axis, the relation

trPS 1 can be represented by a graph below.

We can see from the graph that where we deposit money

without withdrawing or borrow money without paying back,

at the beginning the balance increases slowly and later on it

increases rapidly.

What do you th ink i f n is not equal to 1 ????

Check Understanding

1. Find the total savings for 5,000 baht deposited in an account that pays 5% annualinterest.

2.

Find the total savings for 8,000 baht deposited in an account that pays 2% semiannualinterest.

t0

S

P


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36 CHAPTER 1 Powers

A power is composed of base and exponent such as 25 .

25 is a power of base 5 and exponent 2.

25 25

325 is a power of base 5 and exponent 32 .

32 85 5

It is always mistaken that325 is a power of base 25 and exponent 3. This

is not correct!!!

If we would like to express a power of its base to be a power, we should

put its base inside parentheses, so a power of base 25 and exponent 3

should be 325 or 2 3(5 ) .

325 is a power of base 25 and exponent 3.

3

2 2 3 65 5 5 .

And we will see that32 85 5 but

32 65 5 .

Since 68 55 , so325 is not equal to 325 .

By above information can you find the difference between222

3 and 2223 ?

Unequal


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Mike finds a number of people in his previous nine generation as follows:

First generation 2 people which are parents

Second generations 6 people which are parents and grand parents

Third generations 14 people which are parents, grand parents, and

great grand parents.

Just to here, he confused himself. Can you help him to find that how many

people of his nine generations?

Mikes Family

Documents

Chapter 1 M2 Additional Math