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Powers of Numbers

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If we multiply the same number by itselfseveral times then we can abbreviate theway we write it. For example

2 x 2 x 2 x 2 x 2

can be written as

25 which is read as 2 to the power of 5

Powers of numbers

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The P!"#$T %or multiplication& 5 x 5 ' 25

5 x 5 is called the ()P*+"(" F!, of 25

52 is called the I+"() F!, of 25

In the e-uation 52' 25

5 is called the */( 2 is called the P!0( or I+"()

Notation

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Example

$onsider 21 x 2

in expanded form %2 x 2 x 2& x %2 x 2 x 2 x 2&

which is 23 in index form

$*+ 4!# TI+6 !F * #7( F! ,#7TIP74I+8I+ I+"() F!,9

The Laws of Indices

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Law 1

2a x 2b ' 2a:b

or with a ;eneral base

xa x xb ' xa:b

The Laws of Indices

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Example 2

$onsider 25 < 21

in expanded form %2 x 2 x 2 x 2 x 2& %2 x 2 x 2&

0hich after you cancel down is 22 in index form

$*+ 4!# TI+6 !F * #7( F! "I=I"I+8 I+I+"() F!,9

The Laws of Indices

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Law 2

2a ' 2a>b

2b

or with a ;eneral base

xa ' xa>b

xb

The Laws of Indices

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Example 3

$onsider %1&2

in expanded form % x x & x % x x &

which is ? in index form

$*+ 4!# TI+6 !F * #7( F! P!0(/ !FP!0(/ I+ I+"() F!,9

The Laws of Indices

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Law 3

%2a&b ' 2ab

where ab means a xb

or with a ;eneral base

%xa&b ' xab

The Laws of Indices

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Example 4

$onsider this pattern 2' 2 x 2 x 2 x 2 ' @? 21' 2 x 2 x 2 ' A 22' 2 x 2 ' 2@' 2 2B' 9

$*+ 4!# TI+6 !F * #7( F! P!0( C(! I+I+"() F!,9

The Laws of Indices

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Law 4

2B ' @

or with a ;eneral base

xB' @

*ny number to the power Dero is @

The Laws of Indices

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Example 5 $onsider this pattern 2' 2 x 2 x 2 x 2 ' @? 21' 2 x 2 x 2 ' A

22' 2 x 2 ' 2@' 2 2B' @ 2>@' 999

2>2

' 999

$*+ 4!# TI+6 !F * #7( F! +(8*TI=( P!0(/ I+I+"() F!,9

The Laws of Indices

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Law 5

2>a ' @ 2a

or with a ;eneral base

x>a' @ xa

*ny number to a ne;ative power is one over the number to the positive power

This is called the P!/ITI=( ($IP!$*7

The Laws of Indices

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Law

2@

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232