Download First Index law (multiplying numbers in index form with the same base)

number AND algebra • Number and place value
2
The order is not important when multiplying,
so place numbers first and group the same variables
together.
= 7 ì 3 ì 2 ì m3 ì m8 ì n5 ì n4
3
Simplify by multiplying the numbers and using the
First Index Law for bases that are the same
(add indices).
= 42 ì m3 + 8 ì n5 + 4
= 42m11n9
remember
1. The First Index Law states: am ì an = am + n. This means that when numbers or variables
in index form with the same base are multiplied by each other, the powers (indices) are
added together.
2. If the expression contains coefficients, the coefficients are multiplied together and the
numbers and variables in index form by each other.
3. When there is more than one variable involved in the multiplication question, the First
Index Law is applied to each variable separately.
Exercise
3b
Individual
Pathways
eBook plus
Activity 3-B-1
First Index Law
doc-6839
Activity 3-B-2
More of the First
Index Law
doc-6840
Activity 3-B-3
Advanced use of the
First Index Law
doc-6841
eBook plus
Digital doc
Spreadsheet
Multiplying
with indices
doc-2160
First Index Law (multiplying numbers in
index form with the same base)
Fluency
1 WE 5 Simplify the following after first writing in factor form.
a 22 ì 24 = ( 2 ì 2) ì ( ì ì ì )
=2
b 53 ì 55 = (5 ì 5 ì 5) ì ( ì ì ì ì )
= 5
c f 6 ì f ì f 2 = ( ì ì ì ì ì ) ì ì ( ì )
= f 2 Simplify each of the following.
a 37 ì 32
d 113 ì 113
g 52 ì 52
j q23 ì q24
b
e
h
k
614 ì 63
78 ì 7
89 ì 82
x7 ì x7
c
f
i
l
106 ì 104
211 ì 23
137 ì 138
e ì e3
3 WE 6 Simplify each of the following, giving your answer in index form.
a 34 ì 36 ì 32
b 210 ì 23 ì 25
c 54 ì 54 ì 59
8
2
4
d 6 ì 6 ì 6
e 10 ì 10 ì 10
f 172 ì 174 ì 176
7
8
7
11
10
2
g p ì p ì p
h e ì e ì e
i g15 ì g ì g12
20
12
6
2
10
j e ì e ì e
k 3 ì b ì b ì b
l 5 ì d4 ì d5 ì d7
4 a MC What does 6 ì e3 ì b2 ì b4 ì e equal?
A 6e4b6
B 6e3b6
10
D 6eb
E 6e3b8
C 6eb9
b What does 3 ì f 2 ì f 10 ì 2 ì e3 ì e8 equal?
A 32f 12e11
B 6f 12e11
20
24
D 6f e
E 3f 12e24
C 6fe23
5 WE 7 Simplify each of the following.
a 4p7 ì 5p4
b 2x2 ì 3x6
7
d 3p ì 7p
e 12t3 ì t2 ì 7t
c 8y6 ì 7y4
f 6q2 ì q5 ì 5q8
Chapter 3 Index laws
55
number AND algebra • Number and place value
6 WE 8 Simplify each of the following.
a 2a2 ì 3a4 ì e3 ì e4
b 4p3 ì 2h7 ì h5 ì p3
d 2gh ì 3g2h5
e 5p4q2 ì 6p2q7
8
5
3
4
7
g 9y d ì y d ì 3y d
h 7b3c2 ì 2b6c4 ì 3b5c3
10
2
8
6
20
12
j 10h v ì 2h v ì 3h v
c 2m3 ì 5m2 ì 8m4
f 8u3w ì 3uw2 ì 2u5w4
i 4r2s2 ì 3r6s12 ì 2r8s4
Understanding
7 Simplify each of the following.
a 3x ì 34
b 3y ì 3y + 2
c 32y + 1 ì 34y - 6
8 a Express the following basic numerals in index form: 9, 27 and 81.
b Use your answers to part a to help you
reflection simplify each of the following expressions.
3c
(Give each answer in index form.)
i 34 ì 81 ì 9
ii 27 ì 3n ì 3n - 1
1
2
3
d 3 2 �3 3 �3 4
 
The First Index law can only be
applied if the bases are the same.
Why is that so?
Second Index Law (dividing
numbers in index form with the
same base)
■■
The numbers in index form with the same base can be divided by first being written in factor
26 2 × 2 × 2 × 2 × 2 × 2
form. For example, 26 ÷ 24 = 4 =
2×2×2×2
2
=
2 × 2 × 2 × 2 ×2×2
= 2×2
2×2×2×2
= 22.
■■
The simpler and faster way to divide the numbers in index form is to apply the Second Index
Law. The Second Index Law states: am ó an = am - n. This means that when the numbers
in index form with the same base are divided, the powers are subtracted. For example,
26 ó 24 = 26 - 4 = 22 (as above).
Worked Example 9
Simplify
510
after first writing in factor form, leaving your answer in index form.
53
Think
56
Write
1
Write the problem.
510
53
2
Write in factor form.
=
3
Cancel 5s.
=5ì5ì5ì5ì5ì5ì5
4
Write in index form.
= 57
Maths Quest 8 for the Australian Curriculum
5×5×5×5×5×5×5×5×5×5
5×5×5