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Higher Mathematics

Indices

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Indices Higher Mathematics

What are Indices

Indices are a mathematical shorthand

If, we have: 3 3 3 3 3

We can say that this is: 3 multiplied by itself 5 times

We write this as: 53 pronounced 3 to the power 5

3 is the BASE 5 is the POWER or INDEX


What are Indices

The plural of INDEX is

53 is written in INDEX form

also known as power form

or power notation

INDICES

So,


Rules of Indices


Rules of Indices

3 4a a

Consider the following

What can we make of it ?

Can we generalise to make a rule ?


3 4a a

7a

a a a a a a a 3 times 4 times

a a a a a a a 7 times

3 4 3 4 7a a a a


m na a

m na

.......a a a .......a a a m times n times

.......a a a m + n times

m n m na a a

Generalising


m na a

m na

m n m na a a

When multiplying

We ADD the indices

Generalising


Examples

5 2y y

3p p

Remember – to MULTIPLY you ADD the indices


Rules of Indices




5 2a a


5 2a a

a a a a a a a

5 times

2 times

2 terms on the bottomwill cancel out 2 termson the top


5 2a a

5 2a

a a a a aa a

5 times

2 times

a a a 5 - 2 terms

5 2 5 2 3a a a a

2 terms on the bottomwill cancel out 2 termson the top

3a


m na a

m na

.......a a a .......a a a

m times

n times

.......a a a m - n terms

m n m na a a

n terms on the bottomwill cancel out n termson the top

Generalising


m na a

m na

m n m na a a

When dividing

We SUBTRACT the indices

Generalising


Examples

7

3

y

y

10 4n n

5 2x x

Remember – to DIVIDE you SUBTRACT the indices


Rules of Indices




43a


43a

4 3a

a a a 3 times

4 times

a a a a a a a a a a a a 4 3 terms

3 44 4 3 3 12a a a a

a a a a a a a a a

12a


nma

mna

...a a m times

n times

.......a a a m n terms

...a a ..... ...a a

n mm mn na a a


Examples

Remember – for POWERS you MULTIPLY the indices

23x

23y

31z


Rules of Indices


Can we deduce a meaning from what we know so far ?

What meaning can we give to

0a


0a

0a

a a a 3 times 3 times

m ma

0 1a

a a a a a a

1

3 3a a

a a a

Recall division


Examples

Remember – anything to the power of 0 is 1

8 8p p

5 7 2p p p

3 2 5m m m


Rules of Indices




1a


1a

1a

a a a a a3 times 2 times

3 2a

1a a

a a a a aa

3 2a a Recall division


Examples

Remember – anything to the power of 1 is itself

8 7z z

6 8 3x x x

3 4 6y y y


Rules of Indices




1a


1a

1a


3 4a

1 1

aa

a a a a a a a

1

a



Rules of Indices

a ‘minus’ index means ‘1 over’

A useful way of remembering this

1 1

aa


Rules of Indices




3a


3a

3a


2 5a

33

1a

a

a aa a a a a

3

1

a



na

na

.......a a a .......a a a m times m +n times

( )m m na .......a a a .......a a a a

Generalising m m na a

1nn

aa

1na


Examples

Remember – minus means 1 over

5 7z z

3 7 2x x x

2 3 9y y y

Express in positive index form


Rules of Indices




1

2a


1 1

2 2a a 1aRecall multiplication a

But 2a a a So,

1 1

2 2a a 1

2

2

a

Hence 1

2

2

a a

Thus 1

2a a

1

2a a


Rules of Indices




1

na


1 1

....n na a 1aRecall multiplication a

and,

Hence Thus 1

n na a

n times

1 1

....n na a 1

n

n

a

1

n

n

a a

1

n na a


Examples

Remember – fraction denominator gives the root

1

2x1

3p

Express in root form

1

5z


Rules of Indices




m

na


Recall powers

Thus

nm mna a

So,

1 mm

n na a

But, 1

n na a

1 mm m

nn na a a

m m

nna a


Rules of Indices



Finally, what meaning can we give to

m

na


Examples

Remember – fraction denominator gives the root

2

3x3

5z


3

4p

Remember – fraction numerator gives the power


Recall

Then

and,

1m

nm

na

a

m m

nna a1nn

aa

1m

nm

na

a


Examples

1

2x

2

3z


3

5p

Remember – minus means 1 over


Rules of Indices

Multiplyingm n m na a a

Dividingm n m na a a

Powers n mm mn na a a


Special Indices

Power of 00 1a

Power of 11a a


Negative and Fractional Indices

Negative

1n na a

Fraction m m

n mnna a a

Negative fraction

minus means ‘1 over’

1 1m

nm n mn

aaa


More Examples - 1

Simplify

3xy

43y

52mn

2

3

3a


More Examples - 2

Simplify2 32a a

2 22 3y y

4 212 3x x


More Examples - 3

Multiply out the brackets

2 3y y y


More Examples - 4

Simplify2 3

4

w w

w

3 6

4

t t

t


More Examples - 5

Solve the equation:

2 16n

3 27n


More Examples - 6

Simplify and express in positive index form

13x

33

4u


More Examples - 7

Write in root form 2

3k

1

2x

3

4m


More Examples - 8

Write in index form

3 4z

3 t

1

w

2 x


More Examples - 9

Evaluate1

29

1

38

3

225


More Examples - 10

Simplify3 1

2 2u u

3 1

2 2x x

01

4q


More Examples - 11

Simplify3 2a a

a

1 3

2 2

2

x x

x

3 7 1

4 4 4m m m

Quit

C P Dwww.maths4scotland.co.uk

© CPD 2004

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