Incredible Indices
When we multiply, we
ADD
the powers
4 x  2 x  8x8
3
5
b  3ab  2ab  6a b
2
5
2 8
When we divide, we SUBTRACT the powers
12a  2a 
6
4
6a
2 5
14a b c

3
7ab c
2ab
2
2
Note the c terms cancel out
completely because they have
the same powers!
2 3
a
4a b c

4 7
7 8
3b c
12ab c
2
2
1
5 x yc

6
3
8
2 xc
10 x yc
2
4
3
15x y
3
xy

 3xy
5 xy
1
Remember to place the
letters in the top or bottom
depending on where the
biggest power was !
Remember after you do all
cancelling, and there’s
nothing left in the TOP,
you MUST put a 1.
Remember after you do
3 all cancelling, and there’s
nothing left in the
BOTTOM, you DON’T
have to put a 1.
Anything raised to power zero equals…
1
50
= 1
a0 = 1
( – 356)0 = 1
(2ab2c8)0 = 1
b0 + 7 =
8
3 – t0 = 2
4a0 = 4  x0
=41 =4
a0 + 3b0 + 5c0 = 9
Raising powers to powers using brackets
(a ) means a  a  a which is a
2 3
2
2
2
6
The quick way to get the answer is to
MULTIPLY
the powers!!
(b )  b
5 2
(3b ) 
5 2
(3 y )  27 y
7 3
10
10
9b
21
(6a b )  216a b
5 2 3
15 6
Combined operations 1
3xy 

 2 x3 y 4 

  7 4 5
 3xy  4 x  y 
4 x6 y8 27 x3 y15
 7 20
=
=
2 2
9x y
4x y
2
3x9 y 23
=
x9 y 22
=
3y
5 3
3
4 x6 y8 27 x3 y15
 7 20
2 2
9x y
4x y
Combined operations 2
5
2 xy
3( xy 2 )3
 4x 
 
 3y 
3
5
2
2 xy
9y
 3 6
3x y 16 x 2
8
7
=
3xy
5 6
8x y
=
3y
4
8x
2
Remember with divisions you need to
turn the 2nd fraction upside down!
This is called multiplying by the
reciprocal.
Negative Powers
1
=
a3
1
–
7
b =
7
b
a–3
(3b) – 2
3b – 2
1
1
=
 2
2
(3b)
9b
= 3  b–2 = 3 
a – 3b 4 c – 7 d
b4d
 3 7
ac
1
2
b
3
 2
b
Note – negative power terms go to
the bottom. Others to the top!!
a
 
b
1
When a fraction is raised to a
negative power, you can invert the
fraction and change the power to a
positive!
1
b
 
a
2
 2b 
 3ay 


 
3
ay
2
b




2
9a 2 y 2

4b 2
 2a b 
 4 2 
 5a b 
3 5
3
 5a b 
  3 5 
 2a b 
4 2
3
inverting the stuff in the brackets and
changing the outside power from – 3 to 3
125a12b 6 getting rid of the brackets. 53 is 125, and

3 is 8. Powers are all multiplied by 3
9 15
2
8a b
125a12 a 9

8b15b6
125a 21

8b21
Moving anything with a negative power to
the opposite part of the fraction.
More “exotic” power problems!
1.
8 x  3  2 3 x
41 x 16
(23 ) x 3  23 x

(22 )1 x  24
2 3 x  9  2 3 x
 2 2 x 4
2
2
2 2 x 6
 62 x
2
Recognising all the big
numbers as powers of 2
Removing the brackets. Remember when
you remove brackets, you multiply powers
Multiplying along top & bottom. Remember
when you multiply, you add powers
Now we’ll divide the top by the bottom and this means
we subtract the powers. (2x – 6) – (6 – 2x) = 4x – 12
= 24x – 12 (ans)
Simplify
2.
6 x  3  4 3 x
121 x 18 x 3
Express your answer as powers of 2
and 3
(2  3) x 3  (22 )3 x
 2
(2  3)1 x  (2  32 ) x 3
Breaking all 4 big numbers down
into their prime factors
2 x 3  3 x 3  2 6  2 x
 22 x 1 x x 3 2 x 6 Removing the brackets. Remember when you
remove brackets, you multiply powers
2
3  2 3
2  x  3  3 x 3
Selectively multiplying terms with the same big
 1 x x 5
numbers & adding powers
2 3
Now divide the top by the bottom 2’s: (-x + 3) – ( – 1 – x) = 4
and this means we subtract the
3’s: (x – 3) – ( x – 5 ) = 2
powers (top minus bottom)
= 24 × 32
Or…
144
Fractional powers
1
2
1
3
a  a
1
2
9  9 3
1
2
a  a
1
3
64  64  4
3
a 3 a
1
2
16  16  4
1
2
25  25  5
1
3
8 3 82
100
1
2

1
3
27  3 27  3
1
100
1
2
1
1


100 10
a
Example
m
n
 a
n
8
2
3
m
 8
3
OR
a
m
n

 a
 3 64  4
2
n
m
Using left
formula
OR
8
2
3

 8
3
2
 (2)
2
4
Using right
formula
You can use either of these formulas, but the one on
the right usually avoids getting really big numbers
and so makes the whole process more manageable
Work out 82/3 using
the graphics
Examples using fractional powers
1
6 10 2
(9a b )
=
3 5
3a b
Note that 91/2 = 9 which is 3.
The rest of the powers have been multiplied
6 x ½ = 3 and 10 x ½ = 5
16 x 4 y 8
25( x 6 y 2 )3
First, get rid of any brackets and switch
terms with negative powers to the other part
of the fraction
Note the y – 8 has
been switched
16 x 4

from top to bottom
18 6 8
25 x y y
4x2
 9 3 4
5x y y
2
4x
 9 7
5x y

4
7 7
5x y