﻿ Unit 3 Section 3 : Negative Indices

# 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,

 a–n =

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

 (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,