Download second Index law (dividing numbers in index form with the same base)

number AND algebra • Number and place value
remember
1. The Second Index Law states: am ó an = am - n. This means that when the numbers or
variables in index form with the same base are divided, the powers are subtracted.
2. When the coefficients are present, we divide them as we would divide any other
numbers and then apply the Second Index Law to the variables.
3. If the coefficients do not divide exactly, we simplify the fraction that is formed by them.
4. When there is more than one variable involved in the division question, the Second
Index Law is applied to each variable separately.
Exercise
3c
Individual
Pathways
eBook plus
Activity 3-C-1
Second Index Law
doc-6842
Activity 3-C-2
More of the Second
Index Law
doc-6843
Activity 3-C-3
Advanced use of the
Second Index Law
doc-6844
eBook plus
Digital doc
Spreadsheet
Dividing with
indices
doc-2161
Second Index Law (dividing numbers
in index form with the same base)
Fluency
1 WE 9 Simplify each of the following after first writing in factor form, leaving your answer in
index form.
a
25
22
b
77
73
c
108
10 5
2 WE 10 Simplify each of the following using the index law, leaving your answer in index form.
a 33 ó 32
b 119 ó 112
c 58 ó 54
6
45
42
d 12 ó 12
e 3 ó 3
f 1375 ó 1374
g 623 ó 619
h
1013
10 9
i
15456
15423
h 78
h
k
b 77
b7
l
f 1000
f 100
j
3 WE 11 Simplify each of the following, giving your answer in index form.
a 3x5 ó x3
d 12q34 ó 4q30
b 6y7 ó y5
e 16f 12 ó 2f 3
g 80j15 ó 20j5
h
45 p14
9 p4
i
48g8
6g5
k
81m6
18m 2
l
100 n95
40 n5
j
12b 7
8b
4 a MC What does 21r20 ó 14r10 equal?
A 7r10
2
B 3r
2
2m33
b What does
equal?
16m11
22
8
A m
B
m 22
8
c 8w12 ó w5
f 100h100 ó 10h10
C 7r2
10
D 3r
E
C 8m22
3
D m
E None of the above
2
8
5 Simplify each of the following.
a
15 p12
5 p8
7
d 60 b
20 b
58
Maths Quest 8 for the Australian Curriculum
6
b 18r
c
45a 5
5a 2
10
e 100r
f
9q 2
q
3r 2
5r 6
2 10
r
3
number AND algebra • Number and place value
6 WE 12 Simplify each of the following.
a
8 p6 × 3 p 4
16 p5
b
12b 5 × 4 b 2
18b 2
c
25m12 × 4 n7
15m 2 × 8n
d
27 x 9 y 3
12 xy 2
e
16h 7 k 4
12h6 k
f
12 j8 × 6 f 5
8 j3 × 3 f 2
g
8 p3 × 7r 2 × 2s
6 p × 14r
h
27a 9 × 18b 5 × 4 c 2
18a 4 × 12b 2 × 2c
i
81 f 15 × 25g12 × 16h34
27 f 9 × 15g10 × 12h30
Understanding
7 Simplify each of the following.
a 210 ó 2p
b 27e ó 23e - 4
54 x × 53 y
52 y × 5 x
3 2 − 3 m × 37 m
d
35 m × 3
c
8 Consider the fraction 8 × 16 × 4 .
2 × 32
a Rewrite the fraction, expressing each basic numeral as power of 2.
b Simplify by giving your answer
i in index form ii as a basic numeral.
c Now check your answer by cancelling and evaluating the fraction in the ordinary way.
eBook plus
9 Consider the fraction 6 × 27 × 36 .
12 × 81
a Rewrite the fraction, expressing each
basic numeral as the product of its prime
factors.
b Simplify, giving the answer
i in index form ii as a basic numeral.
Digital doc
WorkSHEET 3.1
doc-6851
3d
reflection  
How will you remember that
when numbers in index form are
divided, powers are subtracted, but
coefficients are divided?
Third Index Law (the power of zero)
■■
Consider the following two different methods of simplifying 23 ÷ 23.
Method 1
2×2×2
2×2×2
8
=
8
=1
23 ÷ 23 =
■■
■■
■■
Method 2
23
23
= 23 − 3 ( using the Second Index Law)
23 ÷ 23 =
= 20
Since the two results should be the same, 20 must equal 1.
Any base that has an index power of 0 is equal to 1.
The Third Index Law states: a0 = 1. This means that any base that is raised to the power of
zero is equal to 1.
If it is in brackets, any numeric or algebraic expression that is raised to the power of zero is
equal to 1. For example (2 ì 3)0 = 1, (2abc2)0 = 1.
Chapter 3 Index laws
59