ML Aggarwal Class 8 Chapter 1 Rational Numbers Exercise 1.3

 Exercise 1.3

Question 1

Sol :

(i)⇒67×1430

=6×147×30

=84210

=25


(ii)  623×127

203×97

20×93×7

=18021

=607


(iii)  259×310

=25×(3)9×10

=7590

=56

=56


Question 2

Sol :

(i) To verify commutative property of multiplication, we have 

to show    45×78=78×45

 L.H.S ⇒ 4×75×8

2840

L.H.S ⇒ 710

R.H.S⇒ 78×45

7×48×5

2840

R.H.S⇒ 710

∴ L.H.S =R.H.S

∴ Commutative property of multiplication is verified

Question 3

Sol :

35×(47×89)=(35×47)×89

L.H.S =35×(47×89)

=35×(4×(8)7×9)

=35×(3263)

=3×325×63

=96315

L.H.S⇒ 32165

 R.H.S =(35×47)×(89)=(3×(4)5×7)×(89)=1235×89=12×(8)35×9=96315R.H.S=32105 

L.H.S=R.H.S, HENCE PROVED

∴This law is called associative property of multiplication

(ii) 59×(32+75)=59×32+59×75


L.H.S =59×(32+75)

=59×((3×5)+(7×2)10) LCM of 2,5=10

=59×(15+1410)

=59×(110)

L.H.S= 118

R.H.S= (59×32)+(59×75)

(5×(3)18]+[5×79×5]

=[5×(3)18]+[5×79×5]

=1518+3545

=(15×5)+(35×2)90

=75+7090

=590

R.H.S =118

L.H.S = R.H,S, Hence proved

This law is called distributive law of multiplication over addition

Question 4

Sol :

(i)  12 reciprocal of 12 =112

∴ 112 is multiplicative  inverse of 12

 ii}  reciprocal of 23=3232 is multipl:cative inverse of 23


(iii)  reciprocal of 47=74 or  74

74 is multiplicative inverse of 47


(iv) 38×(713)=3×(7)8×13=21104

reciprocal of 21104 in 10421

∴ 10421 in multiplicative inverse of 38×(713)


Question 5

Sol :

(i) 25×37114=37×35

=2×(3)5×7114(3×3)7×5

=635114935

(6×2)(1×5)(9×2)70 LCM OF 35,14,35=70

=1251870

=3570

=12


(ii) 89×45+5695×89

=(8×4)9×5+56(9×8)(5×9)

=3245+567245

=(32×2)+(5×15)(72×2)90    L.C.M of 45,6,45= 90

=64+7514490

64+7514490

=590

⇒ 118


(ii) 37×1415×712×(3035)

=3×14×77×15×12×(3035)

=2941260×(3035)

=294×(30)1260×35

15 


Question 6

Sol :

P=827,q=34, r=1215

(i) P(q× r)=(p×q)× r

L.H.S⇒P x (q x r) 

=827×(34×(1215))

=827×(3×(12)4×15)

=827×(3660)

=8×(36)27×60

 LHS=845RHS=(P×q)× r=(827×34)×(1215)=(8×327×4)×(1215)=24108×(1215)=24×12108×15 

R.H.S =845

∴ L.H.S = R.H.S, Hence verified


(ii)

 P×(q r)=P×qP×r.L.HS=P×(q r)=827×(34(1215))=827×(34+1215)=827×((3×15)+(12×4)60)=827×(45+4860)=827×9360=827×3120 

L.H.S =62135

   R.H.S =p×qP× r

827×34(827×(1215)

8×327×4(8×(12)27×15)

2410896405

2932135

(2×15)(32×1)135

=3032135

RHS=62135 

 LHS= RHS; Hence veritied.


Question 7

Sol :

(i) 23×45 is a Rational number

(ii) 5481×63108=63108×5481

(iii) 45×1=45=1×45

(iv) 512×125=1=125×512

(v) 37×(28×59)=(37×28)×59

(vi) 89×[413+517]=(89×413)+(89×517)

(vii) 613×[8947]=613×89(613×47)

(viii) 1625×0=0

(ix) Not defined

 x) 1,1 xi x2 xii) 1 xiii) negative 


Question 8

Sol :

NO,   

= 45×(114)

=45×(54)

=11

             114 is not multiplicative inverse of 45

               multiplicative inverse of 45 should be 54

 

Question 9

Sol :

(i)  {75×(312)}+{75+512}

=75×{312+512} (-i distributive property)

=75×{3+512}

=75×212

=730


(ii) {916×412}+{916×(39)}

916×{412+(39)}( distributive property )

916×{13+(13)}

916{1313}

916×0=0


Question 10

Sol :

Additive inverse of 9=-9

Multiplicative inverse of 9=19

 Required Sum =9+19=81+19=809=889


Question 11

Sol :

Additive inverse of 27=27 

Multiplicative inverse ot 27=72

Required product =27×72

=-1

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