Page 22 - ICSE Math 8

EXERCISE 1.5

1. Simplify.
−15 7 31 2 −8 6 19 7
(a) × (b) × (c) × (d) ×
16 20 11 5 14 −15 − 6 3
2. Multiply and express the result in lowest form.
6 8 − −25 3 −9 5 36 9
(a) and (b) and (c) and (d) and
− 11 10 27 5 11 3 45 12
Properties of Multiplication of Rational Numbers

I. Closure Property: “The product of two rational numbers is always a rational number.”
p r p r
Thus, if and are two rational numbers, then their product × is also a rational number.
q s q s
7
4 − 6  −4 − 
Example 18: Find: (a) × (b)  × 
11 5  5 3 
4 − 6 4×−( 6) −24
Solution: (a) × = = , which is a rational number.
11 5 11 5× 55
7
 −4 −  ( −4) × −7( ) 28
(b)  ×  = = , which is a rational number.
 5 3  53 15
×
Thus, rational numbers are closed under multiplication.
p r
II. Commutative Property: “Two rational numbers can be multiplied in any order.” Thus, if and are
s
p r r p q
two rational numbers, then q × s = s × q .
−   8
 8 −   4 − 
−   4
=
×
×
Example 19: Check if         ?
 9   7   7   9 
−   4
−   8
 8 −  ( −× −8) ( 4) 32  4 −  ( −4) × −8( ) 32
Solution: LHS =     = = and RHS =     = =
×
×
 9   7  97 63  7   9  79 63
×
×
∴ LHS = RHS
Thus, rational numbers are commutative under multiplication.
III. Associative Property: “The three rational numbers can be multiplied in any order.”
p r u  p r  u p r  u 
Thus, if , and are three rational numbers, then  × s  × = ×  × v  .
q s v  q  v q  s 
 2 − 6  4 2  − 6 4 
Example 20: Check if  ×  × = ×  ×  .
 3 7  5 3  7 5 
 2 − 6  4 = { 2×− ( 6) } 4
Solution: LHS =  ×  × ×
 3 7  5 37× 5
−12 4 ( −12) × 4 −48 −16  −48 ( −48) ÷3 −16 
= × = = =   = = 
21 5 21 ×5 105 35  105 105 ÷ 3 35 
2  − 6 4  2 { − ( 6 × ) 4 }
RHS = ×  ×  = ×
3  7 5  3 75×
2  − 24  2×−( 24) − 48 −16
= ×   = = =
3  35  335 105 35
×
∴ LHS = RHS

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