     • date post

17-Jul-2020
• Category

## Documents

• view

6

1

Embed Size (px)

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

• Exponents, Surds

and

Logarithms(Gr 12)

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.

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 

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