Unit 3 Section 2 : Laws of Indices

There are three rules that should be used when working with indices:

When m and n are positive integers,
1. am × an = am + n
2. am ÷ an = am – n   or  
am
an
= am – n   (m ≥ n)
3. (am)n = am × n

These three results are logical consequences of the definition of an , but really need a formal proof. You can 'verify' them with particular examples as below, but this is not a proof:

27 × 23 = (2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= 210 (here m = 7, n = 3 and m + n = 10)

or,

27 ÷ 23 =
2 × 2 × 2 × 2 × 2 × 2 × 2
2 × 2 × 2
= 2 × 2 × 2 × 2
= 24 (again m = 7, n = 3 and m – n = 4)

Also,

(27)3 = 27 × 27 × 27
= 221 (using rule 1)   (again m = 7, n = 3 and m × n = 21)

The proof of the first rule is given below:

Proof

am × an = a × a × ... × a m of these × a × a × ... × a n of these
= a × a × ... × a × a × a × ... × a (m+n) of these
= am+n
The second and third rules can be shown to be true for all positive integers m and n in a similar way.

We can see an important result using rule 2:

xn
xn
= xn – n = x0
but
xn
xn
= 1, so
x0 = 1

This is true for any non-zero value of x, so, for example, 30 = 1, 270 = 1 and 10010 = 1.

Example 1

Fill in the missing numbers in each of the following expressions:

(a) 24 × 26 = 2
(b) 37 × 39 = 3
(c) 36 ÷ 32 = 3
(d) (104)3 = 10

Example 2

Simplify each of the following expressions so that it is in the form an, where n is a number:

(a) a6 × a7
(b)
a4 × a2
a3
(c) (a4)3

Exercises

Question 1
Fill in the missing numbers:
(a) 23 × 27 = 2
(b) 36 × 35 = 3
(c) 37 ÷ 34 = 3
(d) 83 × 84 = 8
(e) (32)5 = 3
(f) (23)6 = 2
(g)
36
32
= 3
(h)
47
42
= 4
Question 2
Fill in the missing numbers:
(a) a3 × a2 = a
(b) b7 ÷ b2 = b
(c) (b2)5 = b
(d) b6 × b4 = b
(e) (z3)9 = z
(f)
q16
q7
= q
Question 3
Explain why 94 = 38.
() = × =
Question 4
Calculate:
(a) 30 + 40
(b) 60 × 70
(c) 80 – 30
(d) 60 + 20 – 40
Question 5
Fill in the missing numbers:
(a) 36 × 3 = 317
(b) 46 × 4 = 411
(c)
a6
a
= a4
(d) (z)6 = z18
(e) (a19) = a95
(f) p16 ÷ p = p7
(g) (p)8 = p40
(h) q13 ÷ q = q
Question 6
Calculate:
(a)
23
22
+30
(b)
34
33
30
(c)
54
52
+
62
6
(d)
77
75
59
57
(e)
108
105
56
53
(f)
417
414
413
411
Question 7
Fill in the missing numbers in each of the following expressions:
(a) 82 = 2
(b) 813 = 9 = 3
(c) 256 = 5
(d) 47 = 2
(e) 1254 = 5
(f) 10006 = 10
(g) 81 = 4
(h) 256 = 4 = 8
Question 8
Fill in the missing numbers in each of the following expressions:
(a) 8 × 4=2 × 2 =2
(b) 25 × 625=5 × 5 =5
(c)
243
9
=
3
3
=3
(d)
128
16
=
2
2
=2
Question 9
Is each of the following statements true or false?
(a) 32 × 22 = 64
(b) 54 × 23 = 107
(c)
68
28
=38
(d)
108
56
=22
Question 10
Complete each expression:
(a) (26 × 23)4 = (2)4 = 2
(b)
()5 = (3)5 = 3
(c)
()4 = (2)4 = 2
(d)
()4 = (3)4 = 3
(e)
()4 = (6)4 = 6
(f)
()5 = (7)5 = 7