Transcript The Rules Of Indices.

The Rules Of Indices.
Rule 1 : Multiplication of Indices.
anxa
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
=…….
What Is An Index Number.
You should know that:
8x8x8x8x8x8= 86
We say“eight 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
4
9
3&2
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:
63x64
= (6 x 6 x 6) x (6 x 6 x 6 x 6)
(1) Expand the expression.
=67
(2) How many 6’s do you
now have?
Key Result.
3
6 x6
4=
6
7
7
(3) Now write the expression
as a single power of 6.
Using the previous example try to simplify the following
expressions:
(1) 3 7 x 3 4
=3
11
(2) 8 5 x 8 9
(3) 4 11 x 4 7 x 4 8
= 8 14
= 4 26
We can now write down our first rule of index numbers:
Rule 1 : Multiplication of Indices.
anxa
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.
What Goes In The Box ? 1
Simplify the expressions below :
(1) 6 4 x 6 3
=67
(2) 9 7 x 9 2
=99
(6) 2 2 x 2 3 x 2 5
= 2 10
(7) 8 7 x 8 10 x 8
(3) 11 6 x 11
(4) 14 9 x 14 12
= 11 7
= 14 21
= 8 18
(8) 5 20 x 5 30 x 5 50
= 5 100
(5) 27 25 x 27 30 = 27 55
Division Of Indices.
Consider the expression:
8 8
7
8
 4
8
7
4
The expression can be
written as a quotient:
Now expand the numerator
and denominator.
8 8 8 8 8 8 8

8 8 8 8
 8 8 8
=8 3
How many eights will
cancel from the top and the
bottom ?
Result:
4
8 7  8 4= 8 3
Cancel and simplify.
Using the previous result simplify the expressions below:
(1) 3 9  3 2
=3
(2) 8 11  8 6
=85
7
(3) 4 24  4 13
= 4 11
We can now write down our second rule of index numbers:
Rule 2 : Division of Indices.
a n a
m
= a n-m
What Goes In The Box ? 2
Simplify the expressions below :
(1) 5 9 5 2
(2) 7
12
7
5
=5 7
=7
7
(6) 2 32  2 27
=25
(7) 8 70  8 39
(3) 19 6  19
(4) 36 15  36 10
(5) 18 40  18 20
= 19 5
= 36 5
= 18 20
= 8 31
(8) 5 200  5 180
= 5 20
Negative Index Numbers.
Simplify the expression below:
5 3 5 7 = 5 - 4
3
5
57
5 5 5

5 5 5 5 5 5 5
1

5 5 5 5
1
 4
5
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:
The result on the previous slide allows us to see the following
results:
Turn the following powers into fractions:
(1)
2
3
(2)
1
 3
2
1

8
3
4
(3)
10
6
1
 4
3
1
 6
10
1

81
1

1000000
We can now write down our third rule of index numbers:
Rule 3 : For negative indices:.
a
-m
1
 m
a
More On Negative Indices.
Simplify the expressions below leaving your answer as a
positive index number each time:
(1)
6
3 3
35
9
6  9  ( 5 )
3
 3 6  9  5
3
8
(2) 7 4  7 3
78  72
7 4  3
 8  ( 2 )
7
1
7
 6
7
7
7
1 6
7
1
 7
7
What Goes In The Box ? 3
Change the expressions below to fractions:
(1)
2
5
(2)
3
3
(3)
1

27
1

32
2
4
23
1

2
2
6
(4)
33
3

4
Simplify the expressions below leaving your answer with a
positive index number at all times:
5
(5)
4 4
43
4
4
6
(6)
7 7  7 6
710  7 11
7
2
2
4
3
3

3

3
(7)
36  3  4  3 2
1
 3
3
Powers Of Indices.
Consider the expression below:
To appreciate this expression
fully do the following:
( 2 3) 2
=(2x2x2 )2
Expand the term inside the
bracket.
Square the contents of the bracket.
= ( 2 x 2 x 2 ) x (2 x 2 x 2 )
=26
Result: ( 2 3 ) 2 = 2 6
Now write the
expression as a power
of 2.
Use the result on the previous slide to simplify the
following expressions:
(1)
( 4 2) 4
=4
(4)
8
(2)
(3)
( 7 5) 4
= 7 20
= 8 42
(3 2) -3
(5) (53 x 54 )4
= 3 -6
5
1
 6
3
( 8 7) 6
 ( 5 1 ) 4
4
We can now write down our fourth rule of index numbers:
Rule 4 : For Powers Of Index Numbers.
( am ) n = a mn
What Goes In The Box ? 4
Simplify the expressions below leaving your answer as a
positive index number.
(1)
4 5
(2)
(7 )
7
20
2
4 3
(
8

8
)
(4)
8
18
(53 )6
1
 18
5
(3)
7 3
(10 )
 10 21
3
2 5
(5) (7  7 )
(6) (116  115 )10
7
 11110
5
Indices & Roots.
This work is covered in Indices 2.