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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
Indices Higher Mathematics
What are Indices
The plural of INDEX is
53 is written in INDEX form
also known as power form
or power notation
INDICES
So,
Indices Higher Mathematics
Rules of Indices
Indices Higher Mathematics
Rules of Indices
3 4a a
Consider the following
What can we make of it ?
Can we generalise to make a rule ?
Indices Higher Mathematics
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
Indices Higher Mathematics
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
Indices Higher Mathematics
m na a
m na
m n m na a a
When multiplying
We ADD the indices
Generalising
Indices Higher Mathematics
Examples
5 2y y
3p p
Remember – to MULTIPLY you ADD the indices
Indices Higher Mathematics
Rules of Indices
Consider the following
What can we make of it ?
Can we generalise to make a rule ?
5 2a a
Indices Higher Mathematics
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
Indices Higher Mathematics
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
Indices Higher Mathematics
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
Indices Higher Mathematics
m na a
m na
m n m na a a
When dividing
We SUBTRACT the indices
Generalising
Indices Higher Mathematics
Examples
7
3
y
y
10 4n n
5 2x x
Remember – to DIVIDE you SUBTRACT the indices
Indices Higher Mathematics
Rules of Indices
Consider the following
What can we make of it ?
Can we generalise to make a rule ?
43a
Indices Higher Mathematics
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
Indices Higher Mathematics
nma
mna
...a a m times
n times
.......a a a m n terms
...a a ..... ...a a
n mm mn na a a
Indices Higher Mathematics
Examples
Remember – for POWERS you MULTIPLY the indices
23x
23y
31z
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
0a
Indices Higher Mathematics
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
Indices Higher Mathematics
Examples
Remember – anything to the power of 0 is 1
8 8p p
5 7 2p p p
3 2 5m m m
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
1a
Indices Higher Mathematics
1a
1a
a a a a a3 times 2 times
3 2a
1a a
a a a a aa
3 2a a Recall division
Indices Higher Mathematics
Examples
Remember – anything to the power of 1 is itself
8 7z z
6 8 3x x x
3 4 6y y y
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
1a
Indices Higher Mathematics
1a
1a
a a a a a a a 3 times 4 times
3 4a
1 1
aa
a a a a a a a
1
a
3 4a a Recall division
Indices Higher Mathematics
Rules of Indices
a ‘minus’ index means ‘1 over’
A useful way of remembering this
1 1
aa
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
3a
Indices Higher Mathematics
3a
3a
a a a a a a a 2 times 5 times
2 5a
33
1a
a
a aa a a a a
3
1
a
2 5a a Recall division
Indices Higher Mathematics
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
Indices Higher Mathematics
Examples
Remember – minus means 1 over
5 7z z
3 7 2x x x
2 3 9y y y
Express in positive index form
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
1
2a
Indices Higher Mathematics
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
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
1
na
Indices Higher Mathematics
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
Indices Higher Mathematics
Examples
Remember – fraction denominator gives the root
1
2x1
3p
Express in root form
1
5z
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
What meaning can we give to
m
na
Indices Higher Mathematics
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
Indices Higher Mathematics
Rules of Indices
What can we make of it ?
Can we deduce a meaning from what we know so far ?
Finally, what meaning can we give to
m
na
Indices Higher Mathematics
Examples
Remember – fraction denominator gives the root
2
3x3
5z
Express in root form
3
4p
Remember – fraction numerator gives the power
Indices Higher Mathematics
Recall
Then
and,
1m
nm
na
a
m m
nna a1nn
aa
1m
nm
na
a
Indices Higher Mathematics
Examples
1
2x
2
3z
Express in root form
3
5p
Remember – minus means 1 over
Indices Higher Mathematics
Rules of Indices
Multiplyingm n m na a a
Dividingm n m na a a
Powers n mm mn na a a
Indices Higher Mathematics
Special Indices
Power of 00 1a
Power of 11a a
Indices Higher Mathematics
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
Indices Higher Mathematics
More Examples - 1
Simplify
3xy
43y
52mn
2
3
3a
Indices Higher Mathematics
More Examples - 2
Simplify2 32a a
2 22 3y y
4 212 3x x
Indices Higher Mathematics
More Examples - 3
Multiply out the brackets
2 3y y y
Indices Higher Mathematics
More Examples - 4
Simplify2 3
4
w w
w
3 6
4
t t
t
Indices Higher Mathematics
More Examples - 5
Solve the equation:
2 16n
3 27n
Indices Higher Mathematics
More Examples - 6
Simplify and express in positive index form
13x
33
4u
Indices Higher Mathematics
More Examples - 7
Write in root form 2
3k
1
2x
3
4m
Indices Higher Mathematics
More Examples - 8
Write in index form
3 4z
3 t
1
w
2 x
Indices Higher Mathematics
More Examples - 9
Evaluate1
29
1
38
3
225
Indices Higher Mathematics
More Examples - 10
Simplify3 1
2 2u u
3 1
2 2x x
01
4q
Indices Higher Mathematics
More Examples - 11
Simplify3 2a a
a
1 3
2 2
2
x x
x
3 7 1
4 4 4m m m
Quit
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