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

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 multiplye.g. 63 definitely does not mean 6 x 3, it means 6 x 6 x 6!

A bit of indices lingo:

45The base

The index or power

Rule 1 – The Multiplication Rule

m n m na a a Using fancy notation:

What it actually means:

Whenever you are multiplying two terms with the same base, you can just add the powers!

Numbers: If there are numbers IN FRONT of your bases, then you must multiply those numbers together as normal

Examples

3 4 7 x x x Classic wrong answer:12 x x√

5 3 8 2 2 2 Classic wrong answer:8 4 x√

94 53 2 6 p p p Classic wrong answer:20 6p x√

4 42 3 222 5 10 ab c ab c a b c √

Remember: if a base does not appear to have a power, the power is a disguised 1!

e.g. 2 1 2 12 2ab c a b c

Rule 2 – The Division Rule

m n m na a a Using fancy notation:

What it actually means:

Whenever you are dividing two terms with the same base, you can just subtract the powers!

Numbers: If there are numbers IN FRONT of your bases, then you must divide those numbers as normal

Examples

812 4 x x x Classic wrong answer:3 x x√

7

34

5

5 5 Classic wrong answer:

4 1 x√

10

55

04

2

5

k

kk Classic wrong answer:

2 4k x√

Or m

m nn

aa

a

Rule 3 – The Power of a Power Rule

( ) m n m na a Using fancy notation:

What it actually means:

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!

Numbers: If there is a number IN FRONT of your base, then you must raise that number to the power

Examples

3 55 1 ( ) x x Classic wrong answer:8 x x√

2 63 (2 ) 2 Classic wrong answer:6 4 x√

3 124 (3 ) 27a a Classic wrong answer:12 9a x√

15 13 2 5 0 5 ( ) 322a b c a b c √

Examples Using all Three Rules

( ) m n m na a Rule 1:

1.

3 2 4

5

( )x x

x

m n m na a a Rule 2: m

m nn

aa

a Rule 3:

3 8

5

x x

x

Rule 3

Rule 1

11

5

x

xRule 2

6x

2.

3 2 2 10

5 2

(5 ) (5 )

(5 ) 5

Rule 3

6 20

10 1

5 5

5 5

Rule 1

26

11

5

5Rule 2

15x

3.

4 2 5 4(5 ) (2 )

50

v v

v

Rule 3

8 2025 16

50

v v

v

Rule 1

28

1

400

50

v

v

Rule 2

278v

Rule 4 – The Zero Index

0 1a Using fancy notation:What it actually means:

Anything to the power of zero is 1!

Examples 0 1 = x 017 = 1 05 5 51 x

Rule 5 – Negative Indices

1

mma

a Using fancy

notation:

What it actually means:

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

Watch out! Only the power and base are flipped over, nothing else!

Examples 22

=1

xx

44

5 = 1

5

1 11 3( ) ( ) 3 1

3 2 2 1 4

( ) ( )4 1

16 3 32 3( ) ( ) 3 2

2

7

8

33

5 =5

aa

Rule 6 – Fractional Indices

n1

na aUsing fancy notation:

What it actually means:

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!

The main ones:

Examples

2

1

a a The power of a half means take the square-root!

31

3 a a The power of a third means take the cube-root!

1

264 64 8 1

3327 27 3

15532 32 2

For ones like the last two it is worth learning your powers of 2 and 3:

Because 33 = 27

Because 25 = 32

2

33

5

6

4

2

4

3 9

3

2 4

2 8

2

27

3 8 16

2

3

2

2 6

1

4

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!

2

38

Examples

1.

Flip it2

31( )8

Root It3

2 2

3

1 1( ) ( )

28

Power It2

2

1

2 4 1

2.

5

61( )64

Flip it

5

664Root It 5 56( 64) 2

Power It 32

Good luck with your revision!