The Rules Of Indices.

1
The Rules Of Indices.
Rule 1 Multiplication of Indices. a n x a m

Rule 2 Division of Indices. a n ? a m .
Rule 4 For Powers Of Index Numbers. ( a m ) n
..
Rule 3 For negative indices a - m .
2
What Is An Index Number.
You should know that
8 x 8 x 8 x 8 x 8 x 8
8 6
We sayeight to the power of 6.
The power of 6 is an index number.
The plural (more than one) of index numbers is
indices.Hence indices are index numbers which are
powers.
The number eight is the base number.
What are the indices in the expressions below
(b) 36 9 34
(c) 8 3 x 7 2
(a) 3 x 5 4
9
3 2
4
3
Multiplication Of Indices.
We know that
7 x 7x 7 x 7 x 7 x 7 x 7 x 7
7 8
But we can also simplify expressions such as
To simplify
6 3 x 6 4
(1) Expand the expression.
(6 x 6 x 6) x (6 x 6 x 6 x 6)
6 7
(2) How many 6s do you now have?
Key Result. 6 3 x 6 4 6 7
7
(3) Now write the expression as a single power of
6.
4
Using the previous example try to simplify the
following expressions
(1) 3 7 x 3 4
(2) 8 5 x 8 9
(3) 4 11 x 4 7 x 4 8
8 14
4 26
3 11
We can now write down our first rule of index
numbers
Rule 1 Multiplication of Indices. a n x a m
a n m
NB This rule only applies to indices with a
common base number. We cannot simplify 3 11 x 4
7 as 3 and 4 are different base numbers.
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What Goes In The Box ? 1
Simplify the expressions below
(1) 6 4 x 6 3
(6) 2 2 x 2 3 x 2 5
6 7
2 10
(2) 9 7 x 9 2
9 9
(7) 8 7 x 8 10 x 8
(3) 11 6 x 11
11 7
8 18
(4) 14 9 x 14 12
(8) 5 20 x 5 30 x 5 50
14 21
5 100
(5) 27 25 x 27 30
27 55
6
Division Of Indices.
Consider the expression
The expression can be written as a quotient
Now expand the numerator and denominator.
How many eights will cancel from the top and the
bottom ?
4
8 3
Result 8 7? 8 4 8 3
Cancel and simplify.
7
Using the previous result simplify the
expressions below
(1) 3 9 ? 3 2
(2) 8 11 ? 8 6
(3) 4 24 ? 4 13
8 5
4 11
3 7
We can now write down our second rule of index
numbers
Rule 2 Division of Indices. a n ? a m a n -
m
8
What Goes In The Box ? 2
Simplify the expressions below
(1) 5 9 ?5 2
5 7
(6) 2 32 ? 2 27
2 5
(2) 7 12 ? 7 5
7 7
(7) 8 70 ? 8 39
(3) 19 6 ? 19
19 5
8 31
(4) 36 15 ? 36 10
(8) 5 200 ? 5 180
36 5
5 20
18 20
(5) 18 40 ? 18 20
9
Negative Index Numbers.
Simplify the expression below
5 3 ?5 7
5 - 4
To understand this result fully consider the
following
Write the original expression again as a quotient
Expand the numerator and the denominator
Cancel out as many fives as possible
Write as a power of five
Now compare the two results
10
The result on the previous slide allows us to see
the following results
Turn the following powers into fractions
We can now write down our third rule of index
numbers
11
More On Negative Indices.
Simplify the expressions below leaving your
answer as a positive index number each time
(2)
12
What Goes In The Box ? 3
Change the expressions below to fractions
Simplify the expressions below leaving your
answer with a positive index number at all times
13
Powers Of Indices.
Consider the expression below
To appreciate this expression fully do the
following
( 2 3 ) 2
Expand the term inside the bracket.
( 2 x 2 x 2 ) 2
Square the contents of the bracket.
Now write the expression as a power of 2.
( 2 x 2 x 2 ) x (2 x 2 x 2 )
2 6
Result ( 2 3 ) 2 2 6
14
Use the result on the previous slide to simplify
the following expressions
7 20
8 42
4 8
3 -6
We can now write down our fourth rule of index
numbers
Rule 4 For Powers Of Index Numbers. ( a m ) n
a m n
15
What Goes In The Box ? 4
Simplify the expressions below leaving your
answer as a positive index number.
16
Indices Roots.
This work is covered in Indices 2.