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Mr F’s Maths Notes
Number
14. Indices
14. Indices
What are Indices?
Indices are just a fancy word for “power”
They are the little numbers or letters that float happily in the air next to a number or letter
A bit of indices lingo:
The base
5
4
The index or power
Two things you must remember about indices…
1. Indices only apply to the number or letter they are to the right of – the base
e.g. in abc2, the squared only applies to the c, and nothing else. If you wanted the squared
to apply to each term, it would need to be written as (abc)2.
2. Indices definitely do not mean multiply
e.g. 63 definitely does not mean 6 x 3, it means 6 x 6 x 6!
Rule 1 – The Multiplication Rule
a m  a n  a mn
Using fancy notation:
Whenever you are multiplying two terms with the same
base, you can just add the powers!
What it actually means:
If there are numbers IN FRONT of your bases, then you must
multiply those numbers together as normal
Numbers:
Examples
x  x
3
4
 x
7
2  2  2
5
3
8
√
Classic wrong answer:
x12
x
√
Classic wrong answer:
48
x
Classic wrong answer:
6p 20
3 p  2 p  6p
4
5
9
√
2ab 2 c  5ab 2c 3  10a 2b 4 c 4
x
√
Remember: if a base does not appear to have a power, the power is a disguised 1!
e.g.
2ab 2 c  2a1b 2 c1
Rule 2 – The Division Rule
a
Using fancy notation:
m
 a
n
 a
mn
Or
am
mn

a
an
Whenever you are dividing two terms with the same base,
you can just subtract the powers!
What it actually means:
If there are numbers IN FRONT of your bases, then you must divide
those numbers as normal
Numbers:
Examples
12
x
 x
4
 x
8
√
57
4
√

5
3
5
20k 10
5

4
k
5
5k
√
Classic wrong answer:
x3
x
Classic wrong answer:
14
x
Classic wrong answer:
4k 2
x
Rule 3 – The Power of a Power Rule
(a )  a
m n
Using fancy notation:
Whenever you have a base and it’s power raised to another
power, you simply multiply the powers together but keep the
base the same!
What it actually means:
Numbers:
mn
If there is a number IN FRONT of your base, then you must raise
that number to the power
Examples
( x5 )3  x15
(23 )2  26
√
Classic wrong answer:
x8
x
√
Classic wrong answer:
46
x
Classic wrong answer:
9a12
(3a )  27a
4 3
12
√
(2a3b2c)5  32a15b10c5
√
x
Examples Using all Three Rules
m
 a n  a mn
Rule 1: a
1.
2.
3.
x3  ( x 2 )4
x5
Rule 3
(53 )2  (52 )10
(55 )2  5
(5v 4 )2  (2v5 ) 4
50v
Rule 2
8v 27
Rule 3
Rule 3
Rule 2:
am
mn

a
an
x 3  x8
x5
56  520
510  51
Rule 1
Rule 1
25v8  16v 20
50v
Rule 3:
x11
x5
526
511
Rule 1
(a m )n  a mn
Rule 2
x6
Rule 2
x15
400v 28
50v1
Rule 4 – The Zero Index
Using fancy notation:
a0  1
What it actually means:
Examples
Anything to the power of zero is 1!
x0 = 1
170 = 1
5 x0  5  1  5
Rule 5 – Negative Indices
Using fancy notation:
What it
actually means:
Watch out!
Examples
a
m
1
 m
a
A negative sign in front of a power is the same as writing “one divided by
the base and power”. The posh name for this is the RECIPROCAL
Only the power and base are flipped over, nothing else!
x
2
1
= 2
x
1
3
( ) 1  ( )1  3
3
1
5
4
1
= 4
5
1 2
4 2
( )  ( )  16
4
1
5a
3
5
= 3
a
2 3
3 3
27
( )  ( ) 
3
2
8
Rule 6 – Fractional Indices
Using fancy notation:
What it
actually means:
The main ones:
a
1
n

n
a
When a power is a fraction it means you take the root of the base… and
which root you take depends on the number on the bottom of the fraction!
1
2
The power of a half means take the square-root!

a
a
1
3

a
3
The power of a third means take the cube-root!
a
Examples
1
2
64
27
32
1
3
1
5
For ones like the last two it is worth
learning your powers of 2 and 3:

64  8

3

5
27  3
32  2
Because 33 = 27
Because
25
= 32
22  4
32  9
23  8
33  27
24  16
34  81
25  32
2 6  64
Flip It, Root It, Power It!
Sometimes you get asked some indices questions that look an absolute nightmare, but if you just
deal with each aspect in turn, then you will be fine:
1. Flip It – If there is a negative sign in front of your power, flip the base over and we’re positive!
2. Root It – If your power is a fraction, then deal with the bottom of it by rooting your base
3. Power It – When all that is sorted, just raise your base to the remaining power and you’re done!
Examples
1.
8

2
3
Power It
2.
1  56
( )
64
Power It
Flip it
12
22
Flip it
32
1 23
( )
8
1

4
64
Root It
5
6
3
1 2
1 2
(3 )  ( )
2
8
Root It
( 6 64)5  25
Good luck with
your revision!