Indices and Factors
Further Index Notation

Indices can also be negative or fractions. The rules below explain how to use these types of indices.

a−1 = This is called the reciprocal of a.
a−n =
a =
a =

Worked Examples

1

Find:

(a)
2−4 Show me
2−4 =
=
=
(b)
3−2 Show me
3−2 =
=
=
(c)
5−1 Show me
5−1 =
(d)
4 Show me
4 =
= 2
(e)
8 Show me
8 =
= 2
(f)
9 Show me
9 = (9)3
= 33
= 3 × 3 × 3
= 27
2

Find

(a)
2−5 × 26 Show me
2−5 × 26 = 2−5+6
= 21
= 2
(b)
m2 × m−4 Show me
m2 × m−4 = m2−4
= m−2
=
(c)
Show me
= 3−7−2
= 3−9
=
(d)
(28 × 26) Show me
(28 × 26) = (28+6)
= (214)
= 214 ×
= 27
(e)
(a2 × b−2)−1 Show me
(a2 × b−2)−1 = a−2 × b2
=
(f)
⎛
⎝⎞
⎠−2 Show me
⎛
⎝⎞
⎠−2		 = (m2a−1)−2
= m−4a2
=

Exercises

Complete the missing numbers, without using a calculator.

(a)
3? = ? =
(b)
2? = ? =
(c)
5? = ? =
(d)
36? = 6 ? =
(e)
36? = ? =
(f)
7? = 49 ? =
(g)
7? = 343 ? =
(h)
17? = ? =
(i)
125? = 5 ? =
(j)
= 2? ? =
(k)
= 2? ? =
(l)
= 10? ? =
(m)
= a? ? =
(n)
= m? ? =
(o)
= p? ? =
(p)
= q? ? =
(q)
= q? ? =
(r)
= q? ? =

Use a calculator to find:

(a)
8−1
(b)
20−1
(c)
⎛
⎝⎞
⎠−1
(d)
⎛
⎝⎞
⎠−1
(e)
15−2 (to 4 d.p.)
(f)
20−3
(g)
81
(h)
243
(i)
16
(j)
144
(k)
169
(l)
121

(a)

Express 81 as a fraction in the form , where a and b are integers.

(b)

Simplify a6 ÷ a2.

(c)

Find the value of y for which 2 × 4y = 64.

y =

Investigation

Find four integers, a, b, c and d such that a3 + b3 + c3 = d3 .