Exponents, Surds and Logarithms(Gr 12) ... Exponents, Surds and Logarithms(Gr 12) Grade 11 CAPS...

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Transcript of Exponents, Surds and Logarithms(Gr 12) ... Exponents, Surds and Logarithms(Gr 12) Grade 11 CAPS...

  • Exponents, Surds

    and

    Logarithms(Gr 12)

    Grade 11

    CAPS

    Mathematics

    Series

  • Outcomes for this Topic

    In this Topic you will:

    • Revise the exponential notation and review the index laws.

    Unit 1.

    • Simplify expressions involving rational exponents.

    Unit 2.

    • Simplify expressions involving surds.

    Unit 3.

    • Revise the logarithmic notation and logarithm laws.

    (mostly done in Grade 12) Unit 4.

  • Grade 11

    CAPS

    Mathematics

    Series

    The

    Exponential

    Notation and

    Index Laws

    Unit 1

  • The Exponential Notation

    42 2 2 2 2   

    We know that :

    ... (to factors of , , )na a a a a n a n a      

    nr a

    4or 2 is the product of 4 factors of 2.

    Exponent

    Base

    Power

  • What if exponents are not positive integers?

      0

    21. 3 1a 

    0 0

    If variable bases are non-zero and is a positive integer then:

    1. 1 (0 is undefined)

    1 2. n

    n

    n

    a

    a a

    2

    2

    1 1 2. 3

    3 9

      

    Examples

  • Index Law 1 (Multiplication)

    Law 1: n m n ma a a   4 5 91. a a a 

    1 22. 2 .2 2n n 

        2 33. 2 . 2 2 8   

          3 2 5

    4. 3 . 3 3 243     

    Examples

  • Index Law 2 (Division)

    Law 2: n m n ma a a   5

    5 3 2

    3 1.

    a a a

    a

     

    6 6 8 2

    8 2

    5 1 1 2. 5 5

    5 5 25

        

    2 1

    2

    2 3.

    10 5 5

    ab a b b

    a b a

     

    Examples

  • Index Law 3 (Exponentiation)

     Law 3: m

    n nma a

      3

    2 2 3 61. a a a 

      2

    5 5 2 102. x x x 

    Examples

  • Index Law 4

    Law 4: ( )m m mab a b

      2

    3 4 3 2 4 2 6 8 81. 2 2 2 64a a a a   

      3

    3 2 3 3 2 3 9 62. 2 3 2 3 2 3     

    Examples

  • Index Law 5

    Law 5:

    m m

    m

    a a

    b b

       

      3

    3 3 3 9

    4 4 3 12

    2 2 2 1.

    3 3 3

        

      2 2

    2 2

    2 2 4 2.

    x x x

        

     

    Examples

  • Example 1: Applying Exponent Laws

    • Simplification using the exponent laws.

     

    6 3

    2 4

    6 .9 1.

    118 . 4

    x x

    x x

       

       

    36 2

    4 2 2 2

    2.3 . 3

    2.3 . 2

    xx

    x x 

    6 6 6

    4 8 4 2

    2 .3 .3

    2 .3 .2

    x x x

    x x x  

    6 4 4 2 6 6 82 .3x x x x x x     4 4 42 .3 16 3x xor 

  • Example 2: Applying Exponent Laws

       

    1 1

    1 1 1

    2 4 2.

    2 2

    n n

    n n n n

     

      

    2

    2

    1 1

    2 2

    2 2

    22

    n n

    nn n

     

      

    2 21 1 2 22n n n n n       2 12

    4

     

    • Simplification using the exponent laws.

    Multiply with the reciprocal

  • Tutorial 1: Simplify Expressions using

    the Exponent Laws

     

    3 2 3

    2

    3 2

    2 2

    4

    2 1

    1

    Simplify the following expressions:

    2 .8 1.

    1 4 .

    4

    12 2 2.

    8 3

    18 2.3 3.

    2

    n n

    n

    x x

    x x

    x x

    x

     

     

         

    PAUSE Unit

    • Do Tutorial 1 • Then View Solutions

  • Tutorial 1 Problem 1: Suggested Solution

    3 2 3

    2

    3 2

    2 .8 1.

    1 4 .

    4

    n n

    n

     

          

       

       

    3 2 3 9

    6 4 4

    2 . 2

    2 . 2

    n n

    n

     

      

    3 2 3 9 6 4 42 n n n      2

  • Tutorial 1 Problem 2: Suggested Solution

    2 2

    4

    12 2 2.

    8 3

    x x

    x x

     

     

     

    2 2 2

    3 4

    3 2 2

    2 3

    x x

    x x

     

      

    2 4 2 3 2 42 .3x x x x x       2 2 92 .3

    4

     

    2 4 2 2

    3 4

    2 .3 .2

    2 .3

    x x x

    x x

      

     

  • Tutorial 1 Problem 3: Suggested Solution

      2

    1

    1

    18 2.3 3.

    2

    x x

    x

    2 1 2 2 22 .3x x x x     72

      2 2 2 2 1

    2 3 2 3

    2

    x x x

    x

      

    3 22 3 

  • Grade 11 CAPS

    Mathematics

    Series

    Unit 2

    Rational

    Exponents

  • What is a rational exponent?

    Rational Number is a real number

    which can be written in the form

    where , and 0 m

    m n n n

     

    We will in this part consider powers

    where the exponent is a rational number.

    We will consider expressions like m

    na

  • Equivalent Notations

    1

    ( , 2, )nna a n n   1

    3 35 5 (From right to left)

    1

    554 4 (From left to right)

  • Equivalent Notations for negative

    rational exponents

    1 1

    1 ( , 2, )nn na a a n n 

        

    1

    3 13 3 1

    5 5 (From left to right) 5

      

    1

    5 1 55 1

    = 4 =4 (From right to left) 4

     

  • Another Equivalent Notation

    2

    3 232 2 (From left to right)

    ( 0, ; , 2) m

    n mna a r a n m n    

    5

    54 43 3 (From right to left) 

     

  • • Apply the exponential laws.

    • Factorize all bases into prime factors.

    21

    341. 81 27 

        1 2

    4 34 33 3 

      1 23 3 

    1 13 3

     

    1 1

    6 42. 125 25 

        1 1

    3 26 45 5 

      1 1

    2 25 5 

      05 1 

    Simplification (Without a calculator) of

    single term exponential expressions

  • • Without variables – simplify each term and add.

          2 3 2

    3 4 33 4 35 2 2 

      

    2 23

    3 341. 125 16 8 

     

    2 3 25 2 2   1 3

    25 8 32 4 4

       

    Example 1: Simplification of polynomial

    exponential expressions

  • • Without variables – simplify each term and add.

    • With variables – factorize.

    12

    2

    9 3 2.

    3

    x

    x

    x

     2 12 2

    3 3 .3

    3 .3

    x

    x

    x

     

    1 3 3 .

    3

    3 .9

    x x

    x

    1 3 1

    3

    3 .9

    x

    x

       

     

    2

    3

    9 

    2 1 2

    3 9 27   

    Example 2: Simplification of polynomial

    exponential expressions

  • • Without variables – simplify each term and add.

    • With variables – factorize.

    2

    1

    3.2 4.2 3.

    2 2

    m m

    m m

     

     

    2 3 16

    2 1 2

    m

    m

     

    13 13

    1

      

    Example 3: Simplification of polynomial

    exponential expressions

  • • Without variables – simplify each term and add.

    • With variables – factorize.

    22 2 6 4.

    2 2

    x x

    x

     

        

    2 2 2 3

    2 2

    x x

    x

      

     

      

      

    2

    2

    Note: Replace 2 with

    2 2 6

    6

    2 3

    2 2 2 3

    x

    x x