Unit 3 Section 3 : Negative Indices

Using negative indices produces fractions. In this section we practice working with negative indices. From our work in the last section, we see that

a2 ÷ a3 = a2 – 3 = a–1

but we know that

a2 ÷ a3 = = , a fraction.

So clearly,

a–1 =

In same way,

a–2 = =
a–3 = =

and, in general,

an =

for positive integer values of n. The three rules at the start of section 3.2 can now be used for any integers m and n, not just for positive values.

Example 1

Calculate, leaving your answers as fractions:

(a) 3–2
(b) 2–1 – 4–1
(c) 5–3

Example 2

Simplify:

(a)
(b) 64 × 6–3
(c) (102)–3

Exercises

Question 1
Write the following numbers as fractions without using any indices:
(a) 4–1
(b) 2–3
(c) 10–3
(d) 7–2
(e) 4–3
(f) 6–2
Question 2
Fill in the missing numbers:
(a)
= =7
(b)
= =10
(c)
= =9
(d)
= =2
(e)
= =10
(f)
= =2
Question 3
Calculate:
(a) 4–1 + 3–1
(b) 6–1 + 2–1
(c) 5–1 – 10–1
(d) 10–2 – 10–3
(e) 4–1 – 10–1
(f) 6–1 + 7–1
Question 4
Simplify the following expressions giving your answers in the form of a number to a power:
(a) 47 × 4–6
(b) 57 × 5–3
(c)
(d) (32)–4
(e) (6–2)–3
(f) 84 × 8–9
(g)
(h)
Question 5
Fill in the missing numbers:
(a)
= 3
(b)
= 10
(c)
= 5
(d)
= 5
(e)
= 6
(f)
= 2
Question 6
Simplify the following expressions:
(a)
(b)
(c)
(d) (x6)–4
(e)
4
(f) (x–8)3
Question 7
Complete the following statements:
(a) 0.1 = 10
(b) 0.25 = 2
(c) 0.0001 = 10
(d) 0.2 = 5
(e) 0.001 = 10
(f) 0.02 = 50
Question 8
Fill in the missing numbers:
(a)
=x2
(b)
x6 × x=x2
(c)
x9 × x=x2
(d)
=x–2
(e)
=x4
(f)
(x3)=x–6
Question 9
Fill in the missing numbers:
(a)
=2
(b)
=5
(c)
=9
(d)
=10
Question 10
If a = b3 and b = , express a as a power of c,
without having any fractions in your final answer.
a =