Lesson 2- Laws of Indices

Objectives

To know what indices areTo learn the rules of indices

Oct 2011 INTO Foundation L2

What are Indices?

Indices provide a way of writing numbers in a more compact and convenient form

Indices is the plural of Index

An Index is often referred to as a power

Oct 2011 INTO Foundation L2

For example

5 x 5 x 5 = 53

2 x 2 x 2 x 2 = 24

7 x 7 x 7x 7 x 7 = 75

7 is the BASE NUMBER

5 is the INDEX orPOWER

75 & 24 are numbers in INDEX FORM

Oct 2011 INTO Foundation L2

75

Combining numbers

5 x 5 x 5 x 2 x 2 x 2 x 2

= 53 x 24

We can not write this any more simply

Can ONLY combine BASE NUMBERS if they are the same

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Rule 1 : Multiplication

26 x 24 = 210

24 x 22 = 26

35 x 37 = 312

General RuleLaw 1

am x an = am+n

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Rule 2 : Division

26 ÷ 24 = 22

25 ÷ 22 = 23

35 ÷ 37 = 3-2

General RuleLaw 2

am ÷ an = am-n

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Rule 3 : Brackets

(26)2 = 26 x 26 = 212

(35)3 = 35 x 35 x 35 = 315

General RuleLaw 3

(am)n = am x n

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Rule 4 : Index of 0

How could you get an answer of 30 ?

35 ÷ 35 = 35-5 = 30

30 = 1 General RuleLaw 4

a0 = 1

Oct 2011 INTO Foundation L2

Putting them together?

25 x 23

24 x 22

= 28

26

26 x 24

23

= 210

23

35 x 37

34

= 312

34

= 27

= 38

= 22

Oct 2011 INTO Foundation L2

Works with algebra too!

a5 x a3

a4 x a6

= a8

a10

= a-2

a6 x a4 = a10

b5 x b7 = b12

c5 x c3

c4

= c8

c4

= c4

Oct 2011 INTO Foundation L2

..and a mixture…

2a3 x 3a4 = 6a7= 2 x 3 x a3 x a4

8a6 ÷ 4a4 = 2a2= (8 ÷ 4) x (a6 ÷ a4)

8a6 4a4

22

Oct 2011 INTO Foundation L2

= 2a2

Fractional indices

(Using Law 1) We could write

2

1

2

11 xxxx

xxx But

So 2

1

xx

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Fractional Indices

Similarly

33

1

333

3

1

3

1

3

1

xxSo

xxxx

xxxx

General RuleLaw 5

nn aa 1

Oct 2011 INTO Foundation L2

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

Oct 2011 INTO Foundation L2

Negative Indices

The last Index rule

am

General RuleLaw 6

a-m = 1

Oct 2011 INTO Foundation L2

SummaryRule 1 : Multiplication of Indices.

a n x a m =………

Rule 2 : Division of Indices.

a n a m = …….Rule 4 : For Powers Of Index Numbers.

( a m ) n = …..

Rule 6 : For negative indices

a - m =……. Rule 3 : For Powers Of Index Numbers.

a 0 = …..Rule 5 : For fractional indices

a1/n = n√aOct 2011 INTO Foundation L2

Exercises

Section 2- Working with Indices

Additional Questions if you get that far!

Oct 2011 INTO Foundation L2

Travelling to Mars

How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?

Oct 2011 INTO Foundation L2

Calculations involving standard form

Time to reach Mars =8.32 × 107

2.6 × 103

= 3.2 × 104 hours

Rearrange speed =distance

timetime =

distance

speedto give

This is 8.32 ÷ 2.6

This is 107 ÷ 103

How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?

Oct 2011 INTO Foundation L2

Calculations involving standard formUse your calculator to work out how long 3.2 × 104 hours is

in years.

You can enter 3.2 × 104 into your calculator using the EXP key:

3 . 2 EXP 4

Divide by 24 to give the equivalent number of days.

Divide by 365 to give the equivalent number of years.

3.2 × 104 hours is over 3½ years.

Oct 2011 INTO Foundation L2