Download Indices (Rules 1)

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Multiplication Rule
am x an = am+n
Division Rule
am  an = am-n
Negative Index Rule
a0 = 1
a-n = 1/an
Write the following as a single exponent and evaluate
25  22
26  22
23  23
3 6  36
23  24
35  36
47  49
23
24
20
30
2-1
3-1
4-2
8
16
1
1
1/2
1/3
1/16
Write the following fractions in index form.
1 1 1 1 1
4 5 6 7 8
4 1 5 1 6 1 7 1 8 1
Write the following as fractional powers.
2
5 2 6 3 7 2 8 1
4
1 1 1 1 1
4 2 52 6 3 7 2 81
The Rules for Indices Division
Consider the following:
35  32 
3x 3x 3x 3x 3
 33
3x 3
24  23 
47  43 
4x 4x 4x 4x 4x 4x 4
 44
4x 4x 4
2x 2x 2x 2
 21
2x 2x 2
For division of numbers in the same base you?
Generalising gives:
Division Rule
subtract the indices
am  an = am-n
Using this convention for indices means that:
5x 5x 5x 5
3x 3x 3x 3
1
2
54  54 
 1  50
34  36 


3
5x 5x 5x 5
3x 3x 3x 3x 3x 3 32
7x 7x 7
73  7 3 
 1  70
1
6x 6x 6
63  67 
 4  64
7x 7x 7
6x 6x 6x 6x 6x 6x 6 6
Generalising gives:
1
n
0 = 1
a

a
In general:
Negative Index Rule
an
and
The Rules for Indices: Multiplication
index 3
base 5
3
5
index 4
4
3
base 3
Consider the following:
32 x 33 = 3 x 3 x 3 x 3 x 3 = 35
(base 3)
24 x 23 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27
(base 2)
53 x 52 x 5 = 5 x 5 x 5 x 5 x 5 x 5 = 56
(base 5)
For multiplication of numbers in the same base you?
Generalising gives:
Multiplication Rule
add the indices
am x an = am+n
Write the following as a single exponent:
23 x 2 5
32 x 3 5
46 x 4 4
53 x 5 1
63 x 6 3
83 x 8 9
2 7 x 22
28
37
410
54
66
812
29
Write the following as a single exponent and evaluate:
22 x 2-3 34 x 3-6 4-4 x 42
2-1
3-2
52  55
63  65
87  89
2 4  28
5-3
6-2
8-2
2-4
4-2
1 1 1 1 1 1 1 1 1 1 1 1 1 1







1
2
2
3
2
2
4
2
2 3
9 4 16 5 125 6 36 8 64 2 16
Write the following as a single exponent and evaluate:
2-3 x 2-2
3-1 x 3-2
4-4 x 43
2-3  22
7-1  7-1
43  4-1
2-5
3-3
4-1
2-5
70 = 1
44 = 256
1
1 1
1 1
1
1 1




5
3
1
5
2
32 3
27 4
32
4 2
The Rules for Indices:
Powers
Consider the following:
(32)3 = 3 x 3 x 3 x 3 x 3 x 3 = 36
(base 3)
(24)2 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 28
(base 2)
(53)3 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 59
(base 5)
To raise an indexed number to a given power you?
Generalising gives:
Power Rule
multiply the indices
(am)n = amn
Write the following as a single exponent:
(22)3
(32)2
(43)4
(53)2
(6-3)2
(8-2)2
(27)-2
26
34
412
56
6-6
8-4
2-14
Indices in Expressions
Simplify each of the following:
1
y2 x y3
y5
2
2y2 x 3y4
6y6
3
5p2 x 3p3 x 2p
30p6
4
8k3 x 2k-4 x 3k2
48k
5
ab2 x a2b3 x a2b4
a5b9
6
2a3b2 x 3ab4 x 2a2b2
12a6b8
7
(2pq2)2
= 2pq2 x 2pq2
= 4p2q4
8
(3a2b3)2
= 3a2b3 x 3a2b3
= 9a4b6
9
(5m2n3)2
= 5m2n3 x 5m2n3
= 25m4n6
10
(2pq2)3
= 2pq2 x 2pq2 x 2pq2
= 8p3q6
11
(3a2b3)4
= 81a8b12
12
(2m2n3)5
= 32m10n15
Raise the number
to the given power
and multiply the
indices.
Simplify the following:
5
1
15 p
3p 1
1
3
2
42
18 p
2
6
p
1
1
4
3
43
54
24m
1 6m 1
 5p 3
 3p 2
4
5
3
43
3p 3

2
3
42
3p 2

2
54
4
15 p
210 p1
18 p
2
12
p1
2
4
 4m
4
6
24m
4m

3
318m 1
Write the following as a power of 2
8 x 16
2 x2
2
(2 ) x (2 )
 5 6  11

5
3
5
2 3
2 x4
2 x2
2
2 x (2 )
4
2
3 4
4 2
12
8
20
2
9
Write the following as a power of 3
9 x 27
(3 ) x (3 )

4
81x 3
34 x 34
2
3
2 2
3 3
3 x3

38
4
9
3
 8  35
3
13
Write the following as a power of 5
5 x 125
2
4
25 x 5
7
3
57 x (53 )3
 2 2 4
(5 ) x 5
16
5
5 x5
8

5
 4 4
8
5
5 x5
7
9