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    Indices

    Curriculum Ready

    Indices

    ACMNA: 209, 210, 212, 264

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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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    Indices

    SERIES TOPIC

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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, its 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

    =

    =

    - - -

  • 4 100% IndicesMathletics 100% 3P Learning

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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= 9

    3=

    2 3- =^ h31 5 =` j

    21 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` j

    Base =

    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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    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 2 242 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 amn mn

    =^ h

    Raising indices to indices

    a

    b

    3 3 352 5 2 10= =#^ h

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

    More Index Laws

    andab a bba

    bam m m mm

    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 24 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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    The Zero Index

    1a aaam mm

    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

    Lets 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 an n1=

    Fractional indices

    36 36 621= =

    8 2x x x8 12 31

    31 12

    31 4#= =#^ h

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    Negative Indices

    Lets 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:

    aa1nn=

    -

    xx

    4 422=

    -

    53

    35

    3

    59252 2

    2

    2

    = = =-

    ` `j j

    aa

    21

    21 1= -

    221

    412

    2- = -=-^

    ^h

    h

    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