Indices and Factors
Index Notation

Index notation is a very useful way of writing expressions like

2 × 2 × 2 × 2 × 2 × 2 × 2

in a shorter format. The above could be written with index notation as 27. The small number, 7, is called the index or power.

Worked Examples

1

Find

(a)
34 Show me 34 = 3 × 3 × 3 × 3 = 81
(b)
45 Show me 45 = 4 × 4 × 4 × 4 × 4 = 1024
(c)
71 Show me 71 = 7
2

Find the missing number.

(a)
34 × 36 = 3? Show me
34 × 36 = (3 × 3 × 3 × 3) × (3 × 3 × 3 × 3 × 3 × 3)
= 310
(b)
42 × 43 = 4? Show me
45 = (4 × 4) × (4 × 4 × 4)
= 45
(c)
= 5? Show me
=
= 5 × 5 × 5
= 53

Note

am × an = am + n and = an − m

These rules apply whenever index notation is used.

Using these rules,

= a3 − 3 = a0 or = = 1

So

a0 = 1
3

Find

(a)
(23)4 Show me
(23)4 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= 212
(b)
(32)3 Show me
(32)3 = (3 × 3) × (3 × 3) × (3 × 3)
= 3 × 3 × 3 × 3 × 3 × 3
= 36

Note

(am)n = am × n

Exercises

Write each of the following using index notation.

(a)
4 × 4 × 4 × 4 × 4
(b)
3 × 3 × 3
(c)
6 × 6 × 6 × 6 × 6 × 6 × 6
(d)
7 × 7 × 7 × 7
(e)
18 × 18 × 18
(f)
19 × 19
(g)
4 × 4 × 4 × 4 × 4 × 4
(h)
7 × 7 × 7 × 7 × 7
(i)
10 × 10 × 10 × 10 × 10 × 10
(j)
100 × 100 × 100 × 100 × 100

Find the value of each of the following.

(a)
34
(b)
54
(c)
74
(d)
104
(e)
50
(f)
36
(g)
27
(h)
21
(i)
84
(j)
41
(k)
30
(l)
52

Fill in the missing numbers.

(a)

27 × 24 = 2

(b)

34 × 35 = 3

(c)

36 × 37 = 3

(d)

4 × 42 = 47

(e)

5 × 52 = 56

(f)

54 × 5 = 59

(g)

2 × 44 = 46

(h)

57 ÷ 54 = 5

(i)

34 ÷ 32 = 3

(j)

714 ÷ 710 = 7

(k)

175 ÷ 17 = 173

(l)

97 ÷ 9 = 93

(m)

46 × 4 = 411

(n)

4 ÷ 46 = 410

(o)

3 × 32 = 38

(p)

36 ÷ 36 =

(q)

37 ÷ 36 =

(r)

30 × 3 = 35

(s)

30 × 37 = 3

(t)

41 × 4 = 48

(u)

52 × 5 = 52

Simplify each of the following, giving your answer in index notation.

(a)
32 × 30 × 34 =
(b)
26 × 27 × 2 =
(c)
52 × 57 × 53 =
(d)
=
(e)
=
(f)
=
(g)
=
(h)
=
(i)
=

Simplify each of the following expressions.

(a)
a3 × a2 =
(b)
a4 × a6 =
(c)
x2 × x7 =
(d)
x4 ÷ x2 =
(e)
y3 × y0 =
(f)
p7 ÷ p4 =
(g)
q6 ÷ q3 =
(h)
x7 × x =
(i)
b4 ÷ b =
(j)
=
(k)
=
(l)
=
(m)
=
(n)
=
(o)
x2 × x3 × x3 =
(p)
=
(q)
=
(r)
=
(s)
=
(t)
=
(u)
=
(v)
(x2)4 =
(w)
(x3)5 =
(x)
(x2 × x7)6 =

243 can be written as 35.

Find the values of p and q in the following:

(a)
64 = 4p p =
(b)
5q = 1 q =

Challenge!

You open a book. Two pages face you. If the product of the two page numbers is 3192, what are the two page numbers?