1. The Rules Of Indices. a n x a m =……… a n  a m = ……. a - m =…….
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 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)
(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
= (6 x 6 x 6) x (6 x 6 x 6 x 6)
(1) Expand the expression.
= 6 7
(2) How many 6’s 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:
(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 x 4 7 as 3 and 4 are different base numbers.

5. What Goes In The Box ? 1 Simplify the expressions below :
= 6 7
(6) 2 2 x 2 3 x 2 5
= 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) 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 ?
=8 3 4
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 6
(3)  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
(1) 5 9 5 2
=5 7
(6)  2 27
= 2 5
(2)  7 5
= 7 7
(7)  8 39
(3)  19
= 19 5
= 8 31
(4)  36 10
= 36 5
(8)  5 180
= 5 20
= 18 20
(5)  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:
(2)
(3)
(1)
We can now write down our third rule of index numbers:
Rule 3 : For negative indices:.
a - m

11. More On Negative Indices.
Simplify the expressions below leaving your answer as a positive index number each time:
(1)
(2)

12. What Goes In The Box ? 3 Change the expressions below to fractions:
(4)
(3)
(2)
(1)
Simplify the expressions below leaving your answer with a positive index number at all times:
(6)
(5)
(7)

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:
(3)
( 8 7 ) 6
(1)
( 4 2 ) 4
(2)
( 7 5 ) 4
= 7 20
= 8 42
= 4 8
(4)
(3 2) -3
(5)
= 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.
(2)
(3)
(1)
(4)
(5)
(6)

16. Indices & Roots.
This work is covered in Indices 2.