This document discusses index notation and powers in mathematics. It provides examples of writing numbers as powers using indices, such as 23 = 2 × 2 × 2. Rules for simplifying expressions with powers are given, such as am × an = am+n. Examples of evaluating powers by substitution are provided, like finding the value of x4 when x = 3.

1. MATHEMATICS

2. Index notation is used to write a number which is
multiplied repeatedly, in a concise way.
2 × 2 × 2 is written as 23 using
indices. That is, 2 × 2 × 2 = 23.
23 is read as “two to the power 3”.
2

3. 3

4. 32 = 3 × 3 = 9
54 = 5 × 5 × 5 × 5 = 625
22 × 3 = 2 × 2 × 3 = 12
62 × 52 = 6 × 6 × 5 × 5 = 900
4

5. 43 74
22 × 32 33 × 52
63
73 3
3
6
216
Seven to the power three
5

6. 16 = 2 × 2 × 2 × 2 = 24
16 = 4 × 4 = 42
6

7. Expressing a number in index notation with a prime
number as the base
7

8. 25 = 5 × 5 = 52
64 = 2 × 2 × 2 × 2 × 2 × 2 = 26
81 = 3 × 3 × 3 × 3 = 34
49 = 7 × 7 = 72
8

9. 18
2
9
3
3
3
1
3
2 24
3
2 6
2 12 15
3
1
3
5
45
5
72
63
1
36
3
18
1
9
3 21
7
7
1
2
3
2
2
3
3
18 = 2 × 3 × 3
= 21 × 32
24 = 2 × 2 × 2 × 3
= 23 × 31 45 = 3 × 3 × 5
= 32 × 51
63 = 3 × 3 × 7
= 32 × 71
72 = 2 × 2 × 2 × 3 × 3
= 23 × 32 9

10. Powers with an algebraic symbol as the base
10

11. 11

12. (i) x4 (ii) a3 (iii) m3 × n3 = m3n3
(iv) 73 × p2 = 73p2
(v) 73 × y4 = 73y4
12

13. a2 = a × a
32x3 = 3 × 3 × x × x × x
23m2 = 2 × 2 × 2 × m × m
2p2 = 2 × p × p
x3y3 = x × x × x × y × y × y 13

14. Finding the value of a power by substitution
14

15. 15

16. (i) x4 = x × x × x × x
= 3 × 3 × 3 × 3
= 81
(ii) 3x2 = 3 × x × x
= 3 × 3 × 3
= 27
16

17. (iii) 5x3 = 5 × x × x × x
= 5 × 3 × 3 × 3
= 135
17

18. (i) 2a2 = 2 × a × a
= 2 × 3 × 3
= 18
(ii) 22a2 = 2 × 2 × a × a
= 2 × 2 × 3 × 3
= 36
18

19. (iii) 7a2 = 7 × a × a
= 7 × 3 × 3
= 63
19

20. (i) x2y3 = x × x × y × y × y
= 1 × 1 × 7 × 7 × 7
= 343
(ii) 2x3y = 2 × x × x × x × y
= 2 × 1 × 1 × 1 × 7
= 14
20

21. (ii) 3xy2 = 3 × x × y × y
= 3 × 1 × 7 × 7
= 147
21

22. (i) a2b = a × a × b
= 2 × 2 × 7
= 28
(ii) ab2 = a × b × b
= 2 × 7 × 7
= 98
22

23. (iii) a3b𝟐 = a × a × a × b × b
= 2 × 2 × 2 × 7 × 7
= 392
23