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					Indices ‘Indices’ is the more mathematical way of saying ‘powers’. You should be familiar with an expression like ‘five squared’ (5²), or ‘ten cubed’ (10³), which can equally be expressed as ‘5 to the power 2’ and ’10 to the power 3’. Well, ‘indices’ is the plural of the word ‘index’ and is used in basically the same way as ‘powers’. In the example ‘5²’, 2 is the index of 5; just as with ‘10³’ – 3 is the index of 10. You can start saying ‘index’ if you want, but ‘power’ is invariably always going to sound way cooler. When you multiply two indices together all you have to do is add the powers. For example: 5 2 x 53 = 52  3 = 55 The reason powers are added like this is because the calculation above can also be written as:
(5x5) - two fives multiplied by (5x5x5) - three fives, which is the same as (5x5x5x5x5) – five fives multiplied together

In general then:

x

a

x

x

b

=

x

a b

Now think about something like (4³)². The ² sign outside the bracket means that whatever is inside the bracket must be multiplied by itself again. So: (4³)² = 4³ x 4³= (4x4x4) x (4x4x4) =

4

6

Another general rule then would be that:

( x a )b = x ab
If the bases are not the same then there are no short cuts and you will have work out the answer. 52 x 32 = 25 x 9 = 225 You may need to use your calculator sometimes. The ‘power’ button usually is xy or just ^. If you are not sure – ask your maths teacher.

To enter 76 into your calculator you would need to press:

xy 7
or

6

^
Whatever indices you multiply, the process is exactly the same; do not tell yourself that it gets any harder, because it really doesn’t. Take the following example: 2/ 4 1/ 4 (2/ 4) (1/ 4) 3/ 4 =6 6 x6 =6 It works just the same way when you’ve got a bracket as well:
(61/ 2 )5/ 4 =

6

5/ 8

…because (1/2) x (5/4) = (5/8)

Fractional powers, like those above, only seem more difficult because they’re harder to see logically; they’re just numbers though, they work in the same way. A fractional power is made up of a root and a power. Looking at the following number triangle you can gather that anything to the power one is just itself: You should know that Compare this with
x x x =

x
=

x

1/ 2

x

x x

1/ 2

=

x

1

x
x is the

You should be able to gather that writing same as writing And so, from the rule So that:
1/ 2

.

( x a )b = x ab if you were given something like
1/ 2

x

3/ 2

it

would be the same as writing ( x

)³ would it not? (1/2) x 3 is (3/2) isn’t it. = ( x )³

x

3/ 2

Conversely

x

2/ 3

is the same as writing ( 3 x )². I’ll let you figure out why.

Once you’re comfortable with writing
3

x

1/ 2

instead of

x or

x

1/ 3

instead of

x you will realise how much more simple and helpful writing fractional powers is.

A final example for this kind of thing is:

When dividing two powers you just subtract them. Take the example of 4 6 /4². You could also write this as:

x

8/ 9

= ( 9 x )8

4 4 4 4 4 4 = 4x4x4x4 = 4 4 = 4 6 2 4 4
Just like before it works the same with fractions as well: 4 5 / 6 / 4 4 / 6 = 4 1/ 6 …because (5/6)-(4/6) = (1/6) In general then:

x

a

/

x

b

=

x

a b

When the power of the numerator is smaller than the power of the denominator then your answer will have a negative power, as shown:

4 4

2 6

=

4 4 1 = = 4 4 4 4 4 4 4 4 4 4

4

26

=

4

4

=

1

4

4

Actually in general:

x

a

is the same as

1

x
x xx
a
a

a

From this information, we can figure out what same as

x

0

is. Well,

xxx
2

2

is the

x

2 2

, which is

x

0

. Indeed,

is also
2 2

x

0

. And so:

x
So

2

x

1

x
x
0

2

=

x

0

=

x x

=1

will always equal 1

Negative powers work in the same way as other powers, all the situations above will also apply to negative powers and negative fractional powers, anything you truly wish. E.g. ( x ) 1/ 3 =
4

x

4 / 3

E.g.
2 3

x/x
8

2

=

x

82

=

x

10

64



2 1 2   1  1  3   1 3   1           64    64    4  16  

2

Key Facts

x a  x b  x a b x a  x b  x a b

x 

a b

 x ab 1 xa

x0  1 xa 
1 a

x ax
Further (more difficult-Higher) Examples

 1 16  16 2    
3 2

3

 
16

3

 43  64

16



3 2



1 64 4   9
 3 2

 1  4   42    1 9  2 9
3 2

3  3 3   4  2  2  8       9   3  33 27    

3



27 8


				
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