Index Laws

What Is An Index Number.You should know that:

8 x 8 x 8 x 8 x 8 x 8 = 8 6 We say“eight to the power of 6”.

The power of 6 is an index number.

The plural of index numbers is indices.

What are the indices in the expressions below:

(a) 3 x 5 4 (b) 36 9 + 34 (c) 3 x 2

4 9 3 & 2

The number eight is the base number.

If the index number is 1 we just write the base number Eg 81=8

Multiplication Of Indices.We know that : 7 x 7x 7 x 7 x 7 x 7 x 7 x 7 = 7 8

But we can also simplify expressions such as :

6 3 x 6 4 To simplify:

(1) Expand the expression.= (6 x 6 x 6) x (6 x 6 x 6 x 6)

(2) How many 6’s do you now have?

7

(3) Now write the expression as a single power of 6.

= 6 7

Key Result.

6 3 x 6 4 = 6 7

We can also simplify expressions such as :

2 x 3 To simplify:

(1) Expand the expression.= ( x ) x ( x x )

(2) How many ’s do you now have?

5

(3) Now write the expression as a single power of .

= 5

Key Result.

2 x 3= 5

Multiplication Of Indices.Multiplication Of Indices.

Using the previous example try to simplify the following expressions:

(1) 3 7 x 3 4

= 3 11

(2) 5 x 9

= 14

(3) p11 x p 7 x p 8

= p 26

We can now write down our first rule of index numbers:

First Index Law: Multiplication of Indices.

a n x a m = a n + m

NB: This rule only applies to indices with a common base number. We cannot simplify 11 x p 7 as and p are different bases.

What Goes In The Box ? 1Simplify the expressions below :

(1) 6 4 x 6 3

(2) g 7 x g 2

(3) d6 x d

(4) 14 9 x 14 12

(5) 25 x 30

(6) 2 2 x 2 3 x 2 5

(7) p 7 x p 10 x p

(8) 5 20 x 5 30 x 5 50

= 6 7

= g 9

= d 7

= 14 21

= 55

= 2 10

= p 18

= 5 100

Division Of Indices.Consider the expression:

47 88 The expression can be written as a quotient:

4

7

8

8 Now expand the numerator

and denominator.

8888

8888888

How many eights will cancel from the top and the bottom ?

4

Cancel and simplify.

888 =8 3

Result:

8 7 8 4= 8 3

Using the previous result simplify the expressions below:

(1) 3 9 3 2

= 3 7

(2) 11 6

= 5

(3) p 24 p 13

= p 11

Second Index Law: Division of Indices.

a n a m = a n - m

We can now write down our second rule of index numbers:

What Goes In The Box ? 2Simplify the expressions below :

(1) 5 9 5 2

(2) p 12 p 5

(3) 19 6 19

(4) 15 10

(5) b 40 b 20

(6) 2 32 2 27

(7) h 70 h 39

(8) 5 200 5 180

=5 7

= p 7

= 19 5

= 5

= b 20

= 2 5

= h 31

= 5 20

Zero IndexConsider the expression:

33 22

3

3

2

2

222

222

8

8

=1 Since the two results should be the same:

2 3 2 3= 20 = 1

Using 2nd Index Law

33 22

3

3

2

2

332 02

Zero IndexUsing the previous result simplify the expressions below:

(1) 30 (2) 6 6

= 0

(3) (p 24 )0

= 1

Third Index Law : Zero Index

a 0 = 1 (where a 0)

We can now write down our third rule of index numbers:

= 1

= 1

Powers Of Indices.Consider the expression below:

( 2 3 ) 2

To appreciate this expression fully do the following:

Expand the term inside the bracket.

= ( 2 x 2 x 2 ) 2Square the contents of the bracket.

= ( 2 x 2 x 2 ) x (2 x 2 x 2 ) Now write the expression as a power of 2.

= 2 6

Result: ( 2 3 ) 2 = 2 6

Use the result on the previous slide to simplify the following expressions:

(1) ( 4 2 ) 4 (2) ( 7 5 ) 4 (3) ( 8 7 ) 6

= 4 8 = 7 20 = 8 42

We can now write down our fourth rule of index numbers:

Fourth Index Law: For Powers Of Index Numbers.

( a m ) n = a m x n

Fifth and Sixth Index Laws

These are really variations of the Fourth Index Law

Fifth Index Law:

(a x b)m = am x bm

Sixth Index Law: m

b

a

mb

ma

Fifth and Sixth Index Laws - Variations

Fifth Index Law:

(2a x 3b)2 = 22a2 x 32b2

= 4a2 x 9b2

= 36a2b2

Sixth Index Law: 2

3

2

b

a

22

22

3

2

b

a

Do not forget to raise the constants to the power as well?Eg:

2

2

9

4

b

a

PracticeMaths Quest 10 Exercice 1A (page 5-6)

Questions 1, 2, 3, 4 & 6: a, b, h, iQuestion 7: a to f

Negative Index Numbers.Simplify the expression below:

5 3 5 7

= 5 - 4 To understand this result fully consider the following:

Write the original expression again as a quotient:

Expand the numerator and the denominator:

5555555

555

7

3

5

5

Cancel out as many fives as possible:

5555

1

Write as a power of five:

Now compare the two results:45

1

The result on the previous slide allows us to see the following results:

Turn the following powers into fractions:

(1) 32

32

1

8

1

(2) 43

43

1

81

1

(3) 610

610

1

1000000

1

We can now write down our seventh rule of index numbers:

For Negative Indices:.

a - mma

1

More On Negative Indices.Simplify the expressions below leaving your answer as a positive index number each time:

(1)5

96

3

33

)5(963 5963

83

(2)28

34

77

77

)2(8

34

7

7

6

1

7

7

617 77

77

1

What Goes In The Box ? 3Change the expressions below to fractions:

Simplify the expressions below leaving your answer with a positive index number at all times:

(1)52 (2)

33

3

2

2

4

(3) (4)

3

2

3

6

3

65

4

44

(5) 1110

67

77

77

(6) (7)246

342

333

333

32

1

27

1

2

1

4

3

44 27 33

1

What Goes In The Box ? 4Simplify the expressions below leaving your answer as a positive index number.

(1) 54 )7(63 )5(

(2) (3)37 )10(

(4)342 )88( (5) 523 )77( (6) 1056 )1111(

207 185

1 2110

188 57 11011

PracticeMaths Quest 10 : Exercise 1B (page 10-12)

Questions 1, 2, 3 : 1st column of eachQuestion 4 : allChallenge questions Question 7 - 11