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Indices

SERIES TOPIC

J 4

Indices is the plural for index. An index is used to write products of numbers or pronumerals easily. For example 42 is actually a shorter way of writing 4 4# . The 2 is the index. Another word for index is exponent.

INDICES

What do I know now that I didn't know before?

Answer these questions, before working through the chapter.

I used to think:

Answer these questions, after working through the chapter.

But now I think:

What is scientific notation?

How are indices used to write very large and very small numbers?

What are indices? (What are exponents?)

What is scientific notation?

How are indices used to write very large and very small numbers?

What are indices? (What are exponents?)

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Basics

Index Notation

Index notation is used to write a product of a number with itself in an easier way. For example:

5 5 5 5 5 6254# # # = =

Index or ‘exponent’ or ‘power’

Basic numeral

Base

So when multiplying a number, say 6, by itself 100 times, it’s easier to write 6100 instead of 6 6 6 ...# # # (100 times).

If the index is 1 we usually make it invisible so we write 7 instead of 71.

Simplifying like terms

Can be simplified (like terms) Cannot be simplified (unlike terms)

Same index

5 5 2 53 3 3+ = ^ h

Same base

Different indices

4 48 6-

Different base

2 43 3+

Adding and Subtracting with Indices

Two expressions in index form are like terms if they have the same base AND they have the same index.

6 4 4 6 43 3 2 3 3+ + + +

Like terms

Like terms

46 6 4 43 3 3 3 2= + + + +^ ^h h

2 2 46 43 3 2= + +^ ^h h

Like terms grouped together

Simplify like terms

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Basics

Same base

Same base

Add indices

Subtract indices

Multiplication with Indices

Consider the product 4 4 4 44 4 4 4 42 3 5 2 3# # ## #= = = +^ ^h h . In the product, the base is the same and the indices have been added together. In general you can apply the formula for multiplication with indices:

a a am n m n# = +

a a am n m n' = -

Multiplying terms with indices

a

a

b

b

4 9 4 9 36 36t t t t t t2 5 2 5 2 5 7# # # #= = =+

Coefficients multiplied separately

Coefficients multiplied separately

Coefficients divided separately

Coefficients divided separately

Same bases grouped together

Same bases grouped together

REMEMBER A coefficient is the number before the variable in an expression. Eg. The coefficient of 2x is 2.

5 3 5 3 15 15p q p q p p q q p q p q3 4 2 3 2 4 3 2 4 1 5 5# # # # # #= = =+ +

Division with Indices

If we divide 7 7 7 7 7 1 7 7 .7 7 7

7 7 7 7 77 7 77 7 75 3 2 2 5 3'

# ## # # #

# ## ## # #= = = = = -c cm m

In the division, the second index (3) has been subtracted from the first (5). In general, apply this formula for division with indices:

Dividing terms with indices

y y y y

y

y

20 4 20 4

5

5

7 2 7 2

7 2

5

' ' '=

=

=

-

^ ^h h

21 7a b c a bc a a b b c c21 75 4 6 4 4 5 4 4 6 4' ' ' ' '= ^ ^ ^ ^h h h h

a b c

ab c

3

3

5 4 4 1 6 4

3 2

=

=

- - -

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BasicsQuestions

1. Write in expanded form:

a

a

d

d

g

g

b

b

e

e

h

h

c

c

f

f

i

52= 64

= 93=

2 3- =^ h31 5

=` j21 4

- =` j

x2= x y3 3

= a b c3 2=

2. Identify the base, index and basic numeral of each of the following:

10

Base =

Index =

Basic numeral =

32

Base =

Index =

Basic numeral =

21 3

` jBase =

Index =

Basic numeral =

1 5-^ h

Base =

Index =

Basic numeral =

26

Base =

Index =

Basic numeral =

3 2-^ h

Base =

Index =

Basic numeral =

1 4^ h

Base =

Index =

Basic numeral =

43

Base =

Index =

Basic numeral =

3. Identify the following values:

a What is the basic numeral of 4 to the index 3?

b What is the base of an expression with index 2 and basic numeral 16?

c What would the index of an expression be if the basic numeral is 81 and the base is 3?

d What is the index in an expression with base 7 and basic numeral 343?

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Questions Basics

4. Use like terms to simplify the following in index form (if possible):

a

a b

c d

e f

g h

i

b

c d

e f

2 23 3+ =

4 5 53 3- =^ h

3 4 6 2 3 62 3 2 3+ + + =^ ^h h

4 2 42 2+ =^ h

7 7 72 2+ + =

3 2 3 4 3 5 3 12 3 2 3- + - + + =^ ^ ^h h h

5. Find the following products in simplest index form:

3 33 6# =

2 23 7#- - =^ ^h h

21

21

213 5

# # =` ` `j j j

x y y x2 4 3# # # =

6 3w v w v4 3 2 8# # # =

5 5 54 3# # =

2 22 5#- - =^ ^h h

qq3 7# =

c a cab 410 2 4 3# =

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BasicsQuestions

6. Find the following products in simplest index form:

a b

c d

e f

g h

i

8 812 5' =

5 511 6'- - =^ ^h h

4

412

19

=

36 9a b a b8 9 5 6' =

24 8e f e f13 6 5 5' =

7 721 15' =

53

539 2

' =c cm m

r s r s5 4 3 2' =

x y

x y

6

547 3

11 4

=

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Knowing More

Raising Indices to Indices

We use what we know about multiplying terms with indices, to find a rule for raising indices to indices. Consider for example:

2 2 2 2 24 2 4 4 8 4 2#= = = #^ h

Notice that the base remains the same and we find the product of the indices. In general, apply the formula

Same base Multiply Indices

a am n mn=^ h

Raising indices to indices

a

b

3 3 35 2 5 2 10= =#^ h

4 4 4x x x3 3 3= =#^ h

More Index Laws

andab a bba

bam m m m

m

m= =^ ch m

If a product or fraction is raised to an index, then the index applies to each term.

Brackets with indices

a b

c d

x x

x

3 3

27

3 3 3

3

=

=

^ h 4

256

256

pq p q

p q

p q

4 2 4 4 4 2 4

4 2 4

4 8

=

=

=

#

^ ^h h

x x

x

2 2

8

3

3

3

3

=

=

` j a b a b

a b

53

5

3

259

3 2

2

2 3 2 2

6 2

=

=

c ^m h

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Knowing More

The Zero Index

1a aaam m

m

m0= = =-

when1 0a a0 !=This means

Anything raised to an index of 0 is 1.

The zero index

4 1 5 1 100000 1 1 1 1 1 1a b x a b5320 0 0 0 0 0 0 3 2 0

= = = = = = - = = = = + = = = = =^ ` ^ ^h j h ha

b

c

a

b

6 2 6 1 2 310 0+ = + =^ ^h h

7 1 7 7ab 0 # #= =^ h

Fractional Indices

Let’s try figure out what to do when the indices are fractions, such as or p1641

71

.

For example, consider 521

:

1

2

3

Using the index law for multiplication we can say 5 5 521 2

1= =^ h

Find the square root of both sides to obtain 5 521 2

=^ h

Simplify by cancelling the index of 2 with the square root 5 521=

For any a, we can say a 21

is the square root of the number a. In the same way a 31

is the cube root of a.

Basically, for any n, a n1

is the nth root of a. Always use the formula

a ann1

=

Fractional indices

36 36 621= =

8 2x x x8 12 31

31 12

31 4#= =#^ h

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Knowing More

Negative Indices

Let’s see if we can use what we know to figure out how to use negative indices such as or3 , x21 10 5- ^ h .

For example:

Negative indices

5 5

55

5

5

5

5

1

55

1

2 0 2

0 2

2

0

2

0

2

2

2`

=

=

=

=

- -

-

-

Since 0 - 2 = -2

Using the division of indices law

According to the zero index law

In general, for negative indices we use the formula:

aa1n

n=-

xx

4 42

2=-

53

35

3

59252 2

2

2

= = =-

` `j j

aa

21

21 1

= -

221

412

2- =-

=-^^

hh

22

1412

2- =- =--

^ h

1t t

t3 333 1 3

3= =-

c m44

16413

3= =-a b

c d

e

f

g

Negative indices examples

In this example, the minus (-) is included in the index.

In this example, it is NOT included in the index.

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Knowing More

More Fractional Indices

Up until now we’ve only worked with fractional indices with numerator 1 like 2521

or b 51

.

As all mathematicians know, these are not the only type of fractions. It is important to learn how to use fractional

indices whose numerators are not 1 for example: 523

Using the raising an index to an index law, we find

4 4 4 453 3

51

35 5 3= = =^ ^h h

In general, the formula for fractional indices is:

a a a a anm m

n nm

mn n m1 1= = = =^ ^ ^h h h

More fractional indices

a b

c d

27

3

81

27

27

34

31 4

3 4

4

=

=

=

=

^

^

h

h

3636

1

36

1

6

1

2161

23

23

3

3

=

=

=

=

-

^ h

1000

x y x y

x y

x y

x y

100 100

100

10

4 623

4 621 3

2 3 3

2 3 3

6 9

=

=

=

=

^ ^

^

^

h h

h

h

8 B 881

81

641

41

32

32

23

3

=

=

=

=

- `

`

j

j

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Questions Knowing More

1. Use the law for raising indices to indices to rewrite the following in simplest index form:

a

a

a

g

d

d

d

j

b

b

b

h

e

e

e

c

c

c

i

f

f

f

32 6=^ h 42

5=^ h

5102=^ h x3

4=^ h

678=^ h

b5 6=^ h

2. Use index laws to rewrite the following in simplest index form:

x2 2 4=^ h

p q10 4=^ h

a b4 2 7 4=^ h

p q r3 3 4 7 4=^ h

32 2

=` j

y

x

2

33

2 3

=e o

y

x3

2 4

=e o

ba

43 3

=` jyx y2 2 4

=c m

yx

32 2

=c m

a b2 7 5=^ h

x y2 4 5 4- =^ h a b c4 2 3 5

=^ h

u3 3 4=^ h

t3 3 4=^ h xy3

5=^ h

3. Use index laws to rewrite the following in simplest index form:

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Knowing MoreQuestions

4. Use the zero index law to simplify the following:

a

a

a

b

b

b

c

c

c

d

d

d

e

e

e

f

f

f

g

g

g

h

h

h

i

i

i

j

j

k

k

l

l

40= 670 = 3622 0

=^ h

50- =

a b c205 7 0=^ h

x3 4 0# =

5 0- =^ h

2 100# =

x2 0# =

3 40=^ h

2 10 0# =^ h

p6 0=

10 2=- 5 1

=- 3 3- =-^ h

4 2- =-

p6 2=-

p q p q20 104 7 6 10' =

4 2- =-^ h

ab 3=-

x x10 103 7' =

3 3- =-

p 7=-

p6 2=-^ h

731=

423=

mn 56= mn 5

6=^ h a b5

274=

q 76= x 4

3=-

10 41=- n 6

1=

5. Simplify the following expressions using the law for negative indices:

6. Use fractional indices to write the following in surd form:

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Questions Knowing More

a b c

a b

c d

e f

g h

i j

d e f

g

8 =

21

1 =

xy

=

831 2

=^ h

w w4 # =

p p21

23

' =

y y35 135' =

x x21 10

03# # # =` j

27 27 61

32 2# =^ ^h h

x x74 34# =

pq370=^ h

216 4261

21

' =^ h

41 2

=-

` j

x =

10

14

=x

15

=

103=

7. Write the following in index notation:

8. Write the following in simplest positive index notation:

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Using Our Knowledge

Area, Volume and Surface Area

Index notation can be applied to measurement, to find expressions for these measurements or to solve problems.

Find an expression for a triangle with the following dimensions:

An expression for the area of this triangle is

Area Base Height 5 2 5x x x x x21

21

21 5 2 2# # ## # # # #= = = =^ ^ ` ^h h j h

This is the plan for a swimming pool:

If the volume of the swimming pool cannot exceed m768 3what is the maximum value for y?

Step 1: Determine an expression for the volume:

Volume length breadth height

y y y

y

6 2

12 3

# #

# #

=

=

=

^ ^ ^h h h

12 768

64

y

y

y

y

64

4

3

3

33 3

=

=

=

=

Step 2: Set the expression equal to 768 and solve for y:

Divide both sides by 12

Find the cube root of both sides

6y2y

y

5x

2x

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Questions Using Our Knowledge

1. A carpet needs to be put down on a stage for an upcoming concert. There is only m490 2 of carpet available for the stage which has the following shape: Find the value of x to ensure all the carpet is used.

4x

3x

x

2. A new building is being built entirely out of glass (including the roof, excluding the floor). The shape of the building is displayed below. If there is only m165 620 2 of glass available, then what is the maximum permissible value of p to ensure there is enough glass for the entire building?

2pp

3p

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Using Our KnowledgeQuestions

3. In an aeroplane, the passenger cabin has dimensions:

Before each flight, the cabin must be filled with oxygen.

Write an expression for the volume of oxygen required for each aeroplane in terms of k.a

b If a particular airline requires a combined volume of 60 000 m3 of oxygen for 10 aeroplanes, then solve for k.

4k6k

3k

2k

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Using Our Knowledge

Scientific Notation for Large Numbers (Greater Than 1)

Scientific notation applies index notation to help write very large and very small numbers in an easy way.

For example,

Some examples are: 5000 5 10

2 500 000 2.5 10

.84 000 000 8 4 10

3

6

7#

#

#

=

=

=

This may seem silly for numbers this ‘small’ but it is certainly helpful for large numbers such as

or3 10 5 1020 100# #

Note that 84 000 000 is not written as 84 106# . Scientific notation requires the first number to be between 0 and 10 (so one of 1, 2, 3, ..., 9).

1000 10 10 10 103# #= =

Scientific notation for large numbers

For large numbers (greater than 1) the index of 10 is the number of digits between the first digit and the decimal point.

. .84000000 0 8 4 107#=First digit

Decimal point Betweem 1 and 10

7 6 5 4 3 2 1

Scientific Notation for Small Numbers (Less Than 1)

Scientific notation is written as (number between 0 and 10) × (index of 10), the only difference for small numbers is that the index is negative. Some examples are:

0.05 5 101005

10

52

2#= = = -

0.0027 27 10 . 10 2.7 1010 00027 2 7 104 4 3## # #= = = =- - -^ h

Again the answer is not written as 27 10 4# since the first number is required to be between 0 and 10.

Scientific notation for small numbers

For small numbers (less than 1) the index is the negative value of the amount of digits after the decimal point up to and including the first nonzero digit.

0.0027 2.7 10 3#= -

1 2 3

Decimal point

First nonzero digit Betweem 1 and 10

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Using Our Knowledge

Scientific Notation Makes Calculations Easier

Using scientific notation

a

c

e

d

b

. .

. .

.

.

.

.

4 2 10 2 5 10

4 2 2 5 10 10

10 5 10

10 5 10

1 05 10 10

1 05 10

7 8

7 8

7 8

15

15

16

# # #

# # #

#

#

# #

#

=

=

=

=

=

=

+

^ ^

^

h h

h

42 000 000 250 000 000#

(Remember - the first number must be between 1 and 10)





.

.

.

.

.

1 5 10 5 10

5 10

1 5 10

51 5

10

10

0 3 10

0 3 10

3 10 10

3 10

8 2

2

8

2

8

8 2

6

1 6

5

# ' #

##

#

#

#

# #

#

=

=

=

=

=

=

=

-

-

^ ^

^

h h

h

150 000 000 500'

0.000443 0.002#

.

4.43 2 10 10

8.86 10

.

4 43 10 2 10

8 86 10

4 3

4 3

4 3

7

# #

#

#

# # #

#

=

=

=

=

- -

- -

- -

-

^ ^h h

.0 00000196

.

.

196 10

196 10

14 10

1 4 10 10

1 4 10

8

21 8

21

4

4

3

#

#

# #

#

=

=

=

=

=

-

-

-

-

-

^ ^

^

h h

h

0.000045 0.000000009'

.

.

.

0.5 10

0.5 10

0.5 10

10

4 5 10 9 10

9 10

4 5 10

94 5

10

10

5 10

5 10

5 9

9

5

9

5

5 9

5 9

4

1 4

3

# ' #

##

#

#

#

#

#

#

#

=

=

=

=

=

=

=

=

- -

-

-

-

-

- - -

- +

-

^ ^

^

^

h h

h

h

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4. Represent the following large numbers using scientific notation:

a

a

a

a

d

d

d

d

b

b

b

b

e

e

e

c

c

c

c

f

f

f

12 000 =

0.0065 =

97 820 000 =

0.00000000004 =

4.5 =

0.0000000476 =

123 000 000 000 =

0.008538 =

780 000 =

0.000723 =

4 050 000 =

0.1 =

5. Represent the following small numbers using scientific notation:

6. Represent the following numbers in scientific notation:

Twelve thousandths

Four hundred and twenty millionths Nine billion, five hundred and sixty seven million

Two million, three hundred thousand

7. Rewrite the following in normal numbers:

.20 34 106#

0.0973 107# .0 00674 10 9# - .0 00004324 104#

.362 983 10 4# - .4387 27 10 8# -

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Using Our KnowledgeQuestions

8. Evaluate the following products using scientific notation:

a

c

e

b

d

f

4 10 7 105 7# # #^ ^h h . .5 2 10 8 3 108 4# # #^ ^h h

. .3 2 10 4 2 107 4# # # -^ ^h h . .6 5 10 1 1 105 3# # # -^ ^h h

. .6 21 10 5 4 1010 12# # # -^ ^h h .3 6 10 5 103 7# # #- -^ ^h h

9. Evaluate the following quotients using scientific notation:

a b

c d

e f

6 10 3 108 4# ' #^ ^h h . .9 6 10 3 2 107 2# ' #^ ^h h

. .8 4 10 2 1 102 4# ' #-^ ^h h .7 2 10 6 103 5# ' #^ ^h h

.6 4 10 8 104 3# ' # -^ ^h h .5 4 10 9 106 4# ' #- -^ ^h h

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10. Evaluate the following using scientific notation:

a

a

b

b

c

c

d

d

e f

64 1012# 100 10100#

64 10123 # 16 10204 #

.1 2 105 2#^ h 8 104 2

#-^ h

11. Evaluate the following using scientific notation:

40 000 650 000# 45 9000'

1210 000 78 0.624'

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Thinking More

Comparing Scientific Notation

Arrange the following numbers in ascending order:

Arrange the following numbers in descending order:

Step 1: Arrange by index

Step 1: Arrange by index

Step 2: Arrange by number in each index

Step 2: Arrange by number in each index

4.5 105# 6.7 104# 4.5 106# 2.6 108# 3.4 105#

. 103 9 3# 8.2 10 3# .5 107 2# . 106 12 5# . 108 2 4#

6.7 104# 4.5 105# 3.4 105# 4.5 106# 2.6 108#

7.5 10 2# - 3.9 10 3# 8.2 10 3# - 8.2 10 4# 6.12 10 5#

6.7 104# 3.4 105# 4.5 105# 4.5 106# 2.6 108#

7.5 10 2# - 8.2 10 3# 3.9 10 3# 8.2 10 4# 6.12 10 5#

Same index

Same index

Prefixes for Indices of 10

We all know that there are 1000 m in 1 km. The prefix kilo actually means 1000 or 103 . So when we say 1 kilometre we mean 1 # 10

3 metres, or when we say 5 kilograms we mean 5 103# grams.

The prefix milli in millimetres means 10 3- (or 0.001). So when we say a distance is 5 mm, we mean 5 10 3# - m (or 0.005 m).

Some other prefixes are:

When we say of 1 terabyte (1Tb), this means 1 1012# bytes of memory.

Prefix Index Abbreviation

pico 10¯12 p

nano 10¯9 n

micro 10¯6 μ

milli 10¯3 m

hecto 102 h

kilo 103 k

mega 106 M

giga 109 G

tera 1012 T

Small number

Large number

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Questions Thinking More

1. Arrange the following in ascending order:

a

a

b

c

b

c

4 105# 3.11 107# 7 106# 3.1 107# 3.2 106#

5 10 4# 4.1 10 4# 5.1 10 3# 1 10 1# 2 10 3#

7.2 10 4# 6 102# 7.1 10 3# 3 2 103#

2. Complete the following table:

Measurement Prefix Index Small or Large Number Expanded Form

100 Mb Mega 106 Large 100 000 000 b

65 nm Nano

97 μm

640 pg 10-12

102 Large 200 m

3 000 000°C

0.000 000 004 m

The mass of the Earth is estimated to be 5.98 1024# kg. Write this value in teragrams (Tg).

The distance from the Earth to the Sun is estimated at 1.496 1011# m. Write this in Gigametres (Gm).

The charge of an electron is 1.60219 10 19# - Coulombs (C). Write this in picocoulombs (pC).

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Thinking Even More

Finding Missing Terms

Find d in each of the following examples:

Finding the missing term in a product

To obtain d by itself, divide both sides by everything other than d:

g g

g

g

g

g

g

g

4 7

4

4

4

7

7 4

3

#

#

=

=

=

=

-

4

4

4

4

Finding the missing term in a product

To obtain d by itself, divide both sides by everything other than d:

3 12

4

12

v w v w

v w

v w

v w

v w

v w

vw

3

3

3

312

2 3 3

2

2

2

3 3

3 2 3 1

2

#

#

=

=

=

=

- -

4

4

4

4

Finding the missing numerator in a quotient

To obtain d by itself, multiply both sides by the denominator under d:

6 2

12

aa

aa a a

a a

a

26

22 6 2

3

5

3

3 5 3

5 3

8

# #

#

=

=

=

=

4

4

4

4

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J 4

Thinking Even More

Finding Missing Terms

Finding the missing denominator in a quotient

To obtain d by itself, swap the denominator with the term on the other side of the equals sign:

7

b c b c

b c

b c

b c

b c

21 3

3

21

321

8 32

2

8 3

8 2 3 1

6 2

=

=

=

=

- -

4

4

4

4

Finding the missing value in a surd

To obtain d by itself, raise both sides to the index which will cancel the surd away:

8

a b

a b

a b

2

2

3 2 4

3 3 2 4 3

6 12

=

=

=

4

4

4

^ ^h h

Finding the missing value in an index

To obtain d by itself, find the square root of both sides to cancel the square away (assume d is positive).

a b

a b

a b

9

9

3 6

2 8 12

2 8 12

4

=

=

=

4

4

4

^

^

h

h

(Remember the square root is positive by convention)

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Thinking Even MoreQuestions

1. Find the value of d in the following products:

a

c

e

g

b

d

f

h

r r3 5# =4 2 8a a2 5# =4

5m n m n3 2 6 8# =4 8 24b a b2 2 3# =4

4 20x yz yz x4 3 6 8# =4 4 16p q r p q r3 4 2 5 6 7# =4

3 15a b c abc a b c2 3 5 8 7# # =4 4 2 32d e f d e f d e f4 5 2 2 6 4 10 16 9# # =4

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SERIES TOPIC

J 4

Questions Thinking Even More

a

a

c

c

b

b

d

d

2. Find the value of d in the following quotients:

3y

y30 6

2=

49

t ut u

5 2 3

6 5=4

6 6p q p q4 2 5 8' =4 54 6x y z x y8 10 5 4 3' =4

3. Find the value of d in the following:

4xy z2=4 3a b3 3 5=4

16p q4 8 12=4^ h 64m n p

3 15 9 21=4^ h

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SERIES TOPIC

J 4

Thinking Even MoreQuestions

4. Questions to think about:

a

b

c

d

e

f

g

h

Explain the difference between 3 4 3#^ h and 3 43# .

What is the difference between ab 0^ h and ?ab0

What is the difference between 46 and 4 106# ?

For any number a, does a0 exist? If so then what is its value, if not then why not?

Does 1031

exist? If so then what is its value, if not then why not?

Is there a value for d so that 5 35x x10 4# =4 ?

Do 60and 6

10

have different values?

What are the values of x and y if 81a b a b3 y x3 16 12=^ h ?

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J 4

Answers

Basics: Basics:

1. 2.

3.

4.

5.

6.

2.

a

d

g

b

e

h

c

f

i

5 5 52 #=

6 6 6 6 64 # # #=

9 9 9 93 # #=

2 2 2 23 # #- =- - -^ h

31

31

31

31

31

315

# # # #=` j

21

21

21

21

214

# # #- = - - - -` ` ` ` `j j j j j

x x x2 #=

x y x x x y y y3 3 # # # # #=

a b c a a a b b c3 2 # # # # #=

a

b

c

Base = 10

Index = 1

Basic numeral = 10

Base = 3

Index = 2

Basic numeral = 9

Base = 4

Index = 3

Basic numeral = 64

Base = 21

Index = 3

Basic numeral = 81

Base = -3

Index = 2

Basic numeral = 9

Base = 2

Index = 6

Basic numeral = 64

f Base = 1

Index = 4

Basic numeral = 1

g

a

a b

c d

e

f

b

c

d

h

Base = -1

Index = 5

Basic numeral = -1

so base is( )4 4=4

4=4 (Index is 4)

3=4 (Index is 3)

64

a b

c d

e f

g h

i

2 23^ h

3 53^ h

3 53 62 3+^ ^h h

3 42^ h

7 2 72+ ^ h

5 7 13 32 3- + +^ ^h h

39

2 10-^ h

21 9

` j

x y5 5

w v18 6 11

58

2 7-^ h

q10

40a b c4 2 5

a b

c d

e f

g h

i

87

5 5-^ h

47

a b4 3 3

e f3 8

76

53 7

` j

r s2 2

x y9 4

d

e

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SERIES TOPIC

J 4

Answers

a

a

g

d

d

j

b

b

h

e

e

c

c

i

f

f

312 410

656 520

x12 b30

x24 8 t34 12

x y5 15 p q40 4

a b10 35 u34 12

a b44 8 28 x y2 4 16 20-^ h

a b c20 10 15 p q r34 12 16 28

a

d

b

e

c

f

3

22

2

y

x12

8

y

x

3

22 2

2 2

y

x

2

33 9

3 6

b

a

4

33 3

3 3

x24 8

Knowing More: Knowing More:

Using Our Knowledge:

1. 6.

7.

8.

1.

2.

3.

4.

2.

3.

4.

5.

a b c

d e f

g h i

j k l

1 1

1

1 1

1

3 1-

2

26 12

a b

c d

e f

g h

1001

51

271-

271-

161

161-

p

17 b

a3

a b

c d

e f

g h

i

i j

k l

p

62 p36

12

x

14 p q

22 3

73

10

14

n6 42 3

q67

x

1

43

m n65 m n6 65

a b25 47

a b

c

f

g

821

x 21

1031

d 21 21-

e 10 41- x 5

1-

x y1

a b

c d

e f

g h

i j

4 16

w 29

27

p1 x 4

10

y

12

1

x1031 2 3

mx 7=

mp 91=

a

b

Total volume k48 3=

mk 5=

a b.1 2 104# .7 8 105#

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SERIES TOPIC

J 4

Answers

Using Our Knowledge: Thinking More:

4. 1.

2.

5.

6.

7.

8.

9.

10.

11.

a

d

d

b

e

e

c

c

f

f

.4 05 106# .9 782 107#

.4 5 100# .1 23 1011#

.6 5 10 3# - .7 23 10 3# -

10 1- 4 10 11# -

.4 76 10 8# - .8 358 10 3# -

a

a

d

d

b

b

e

c

c

f

.1 2 10 2# -

.4 2 10 4# -

.2 3 106#

.9 567 109#

.2 034 107# .3 62983 10 2# -

.4 38727 10 5# - .9 73 105#

6.74 10 12# - .4 324 10 1# -

a

c

e

e

b

d

d

f

f

.2 8 1013# .4 316 1013#

.1 34 104# .7 15 102#

.3 3534 10 1# - .1 8 10 9# -

a b2 104# 3 105#

c 4 10 6# - .1 2 108#

8 106# 6 10 3# -

a b

b

c

c

d

d

8 106# 1051

4 104# 2 105#

e f.1 44 1010# .1 5625 10 10# -

a .2 6 1010# 5 10 3# -

.1 1 103# .1 25 102#

a

a

b

c

b

c

4 105# , 3.2 106# , 7 106# ,

3.1 107# , 3.11 107#

4.1 10 4# , 5 10 4# , 2 10 3# ,

5.1 10 3# , 1 10 1#

7.2 10 4# , 7.1 10 3# , 3 ,

6 102# , 2 103#

Mea

sure

men

tPr

efix

Inde

xSm

all

or L

arge

N

umbe

rEx

pand

ed F

orm

100 M

bM

ega

10

6La

rge

100 0

00 0

00 b

65 n

mN

ano

10

-9

Smal

l0.0

00 0

00 0

65

m

97 μ

mm

icro

10

-6

Smal

l0.0

00 0

97

m

640 p

gpi

co10

-12

Smal

l0.0

00 0

00 0

00 6

4 g

2 h

mhe

cto

10

2La

rge

200 m

3 M

Ccm

ega

10

6La

rge

3 0

00 0

00

°C

4 n

mna

no10

-9

Smal

l0.0

00 0

00 0

04 m

Tg.5 98 1015#

Gm.1 496 102#

pC.1 60219 10 7# -

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SERIES TOPIC

J 4

Answers

a

c

e

g

b

d

f

h

b

d

5x z4 3=4

r2=4 a4 3=4

m n5 3 6=4 a b3 2

=4

p q r4 2 2 5=4

d e f4 4 5 3=4a b c5 2 4 5

=4

Thinking Even More: Thinking Even More:

1.

2.

3.

4.

4.

a

a

c

c

b

d

t u45 8 8=4

16x y z2 4 2=4

y10 4=4

36p q9 10=4 x y z9 4 7 5

=4

a b27 9 15=4

2p q2 4=4 4m n p5 3 7

=4

a

b

c

d

e

f

g

The first is cubing (raising to the index of 3) the product of 3 and 4, that is 12 1443

= .

The second is taking the product of 3 and 4 cubed, that is 3 64 192# = .

The first is raising the product of a and b to the index of zero, that is ab 10

=^ h .

The second is multiplying a by b0 , that is a b a0# = .

The first is a short hand way of writing 4 4 4 4 4 4# # # # # , which is equal to 4096.

The second is a short hand way of writing 4 10 10 10 10 10 10# # # # # # , which is equal to 4 000 000.

We can usually write any surd like asa ann1

But if we do this with a0 we would get a 01

and so on. Since we can’t divide a number by zero it tells us that a0 does not exist.

10 100031

=

h

Yes, there is a value for d that makes 5 35x x10 4# =4 . The value is x7 6

No, the two terms do not have different values; they are both equal to 1.

x = 4 and y = 4

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Indices