1. an is a short hand way of writing
Indices
an is a short hand way of writing
a x a x a ……. (n factors)
a is called the base number
and n is called the index number
Calculate : 23 x 22
2 x 2 x 2
x 2 x 2
= 32 25
= 32
Can you spot the connection ?

2. am x an = a(m + n) simply add powers
Indices
4 x 4 x 4 x 4 x 4
Calculate : 45 ÷ 42 =
= 43
4 x 4
Can you spot the connection !
am x an = a(m + n) simply add powers
am ÷ an = a(m - n) simply subtract powers

3. am x an = a(m + n) simply add powers
Some Examples
am x an = a(m + n) simply add powers
(5) t 5 x t 3
= t 8
(1) a 4 x a 3
= a 7
(6) q2 x q3 x q5
= q 10
(2) p 7 x p 2
= p 9
(7) h 7 x h 10 x h
= h 18
(3) x 6 x x
= x 7
(8) g 2 x g 3 x g 5
= g 10
(4) b 9 x b 12
= b 21

4. am ÷ an = a(m - n) simply subtract powers
Some more examples
am ÷ an = a(m - n) simply subtract powers
(1) r 9  r 2
(5) c 9  c 2
= c 7
= r 7
(2) m 12  m 5
= m 7
(6) d 10  d 7
= d 3
(3) k 6  k2
= k 4
(7) s 12  s 6
= s 6
(4) y 15  y 10
= y 5
(8) n 11  n 3
= n 8

5. am ÷ an = a(m - n) simply subtract powers
Fractions as Indices
More Rules a3 a5 = = 1 a2
a x a x a
a x a x a x a x a
By the division rule
am ÷ an = a(m - n) simply subtract powers 1 an =
a –n a3 a5 = 1 a2 =
a -2
a 3 - 5
= a -2

6. am ÷ an = a(m - n) simply subtract powers
Fractions as Indices
More Rules a5 =
a x a x a x a x a
= 1
a x a x a x a x a
am ÷ an = a(m - n) simply subtract powers a5 =
a 5 - 5
= a 0
a0 = 1

7. Fractions as Indices (am)n = amn More Rules (a5)3 = a5 x a5 x a5 =
a3 x a3 x a3 x a3 x a3
= a
= a15
(am)n = amn

8. Fractions as Indices
a½ x a½
= a½ + ½
= a √ a x
= a √ a
= a½

9. √ √ a⅓ x a⅓ x a⅓ = a⅓ + ⅓ + ⅓ = a a = a a = a⅓ Fractions as Indices x3
= a √ a
= a⅓ 3

10. Fractions as Indices
In general we have

11. Final Rule

12. Examples : Simplify the following

13. Examples
(am)n = amn

14. Example