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

but we know that

So clearly,

In same way,

and, in general,

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:

Example 2

Simplify:

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 =