ML Aggarwal Solution Class 9 Chapter 8 Indices Exercise 8

 Exercise 8

Question 1

(i) (8116)34

(3424)3/4

[(32)4]3/4

(32)4×34

(32)3

(23)3

2333

2333

2×2×23×3×3

827


(ii) (16164)2/3

(12564)2/3

(5343)2/3

[(54)3]2/3

(54)3×23

(54)2

(45)2

4252

1625


Question 2

(i) (2a-3b2)3

⇒23a-3×3b2×3

⇒8a-9b6


(ii) a1+b1(ab)1

1a+1b1ab

b+aab1ab

b+aab×ab1

⇒a+b


Question 3

(i) x1y1x1+y1

(xy)11x+1y

1xyy+xxy

1x+y


(ii) (4×107)(6×105)8×1010

4×107×6×1058×1010

3×10751010

3×1021010

310102

3108


Question 4

(i) 3ab1+2ba1

Sol :

3a(1b)+2b(1a)

⇒3ab+2ab

⇒5ab


(ii) 50×4-1+81/3

1×14+(23)13

14+2

1+84

94


Question 5

(i) (8125)1/3

(2353)1/3

(25)3×13

(25)1

52


(ii) (0.027)13

(271000)1/3

(33103)1/3

(310)8×1/8

(310)3×13

103


Question 6

(i) (127)2/3

(133)2/3

1(33)2/3

133×23

132

⇒-32

⇒-9


(ii) (64)-2/3 ÷ 9-3/2

⇒(43)-2/3÷(92)-3/2

43×23÷32×37

⇒4-2÷3-3

4233

142133

3342

2716


Question 7

(i) (27)2n/3×8n/6(18)n/2

(33)2n3×(23)n/6(2×9)n/2

33×2n3×23×n62n2×32×n2

32n×2n/22n/2×3n

32n2n/2×12n/2×3n

32n+n2n2×2n2

33n2n/22n/2

33n×2n/22n/2

⇒33n


(ii) 5(25)n+125(5)2n5(5)2n+3(25)n+1

51(52)n+15252n5152n+3(52)n+1

51+2n+252+2n51+2n+352n+2

52n+352+2n52n+452+2n

52n+1535252n52n545252n

52h[5352]53n[5452]

1252562525

100600

16


Question 8

(i) (843÷22)12

((23)43÷22)1/2

(2422)1/2

(24+2)1/2

(22)1/2

⇒2-1

12


(ii) (278)2/3(14)2+5

(3323)23(122)2+1

(32)3×23122×2+1

(32)2124+1

9424+1

9416+1

9415

9604

514


Question 9

(i) (3x2)3×(x9)23

Sol :

1(3x2)3×x9×23

133x2×3×x3×2

127x6×x6

127


(ii) (8x4)1/3÷x1/3

(23x4)1/3÷x1/3

2313x413x13

2x4/3x1/3

2x4×13x13

x13[2x41]


Question 10

(i) (32)0+34×36+(13)2

30+34+6+132

⇒30+3-4+6+32

1+132+32

⇒1+9+9

⇒19


(ii) 9523(5)0(181)12

9523(1)(192)12

9523192×12

32×523191

⇒35-3-9

⇒35-3-32

⇒3(34-1-3)

⇒3(81-1-3)

⇒3(77)

⇒231


Question 11

(i) 1634+2(12)130

(24)34+2(121)1

24×34+22

⇒23+4

⇒8+4

⇒12


(ii) (81)34(132)25+(8)13(12)1(2)0

(34)34(125)2/5+(23)13(121)

33125×25+2(2)

33122+2(2)

⇒27-22+4

⇒27-4+4

⇒27


Question 12

(i) (64125)23÷1(256625)14+(25364)0

(4353)2/3÷1(4454)14+1

(45)3×23÷1(45)4×14+1

(45)2÷1(45)+1

(54)2(15)+1

(54)2×(45)+1

54+1

5+44

94


(ii) 5n+36×5n+19×5n22×5n

5n536×5n59×5n22×5n

5n[536×5]5n[94]

125305

955

⇒19


Question 13

(i) [(64)2322÷80]12

((43)23122÷1)1/2

(4222)1/2

(42)2×12

⇒2-1

12


(ii) 3n×9n+1 ÷ 3n-1×9n-1

⇒3n×32(n+1) ÷ 3n-1×32(n-1)

⇒3n×32n+2 ÷ 3n-1×32n-2

3n+2n+23n1+2n2

33n+233n3

33n3233n33

⇒32×3+3

⇒32+3

⇒35

⇒243


Question 14

(i) 22×4256364(12)2

(22)1/2×(44)1/4(43)13122

2×4422

⇒2-4

⇒-2


(ii) 367×437×937×26722+20+22

367×3237×22×37×2674+1+122

367×367×267×2674+1+14

367+67×267+6716+4+14

30×20(214)

1(214)

421


Question 15

(i) (32)25×(4)12×(8)1322÷(64)13

(25)25×(27)12×(23)13122÷(43)13

22×21×21122÷(41)

212+1(122)(122)

⇒2-2

122

14


(ii) 52(x+6)×257+2x(125)2x

52x+12×52(7+2x)(53)2x

52x+12×514+4x56x

52x+1214+4x56x

56x256x

56x5256x

⇒5-2

152

125


Question 16

(i) 72n+349n+2((343)n+1)2/3

72n+372(n+2)(73(n+1))2/3

72n+372n+472(n+1)

72n+372n+472n+2

72n7372n7472nn2

22x[7374]72n72

343240149

205849

⇒-42


(ii) (27)4/3+(32)0.8+(0.8)1

(33)43+(25)810+(810)1

34+25×45+(45)1

34+24+54

81+16+54

97+54

388+54

3934


Question 17

(i) (325)13(32+5)13

[(255)(25+5)]13

[(25)2(5)2]13

[(25)2(5)2]13

(255)13

(325)13

(27)13

(33)13

⇒3


(ii) (x13x13)(x23+1+x23)

(x1/31x1/3)(x2/3+1+1x2/3)

∵It is in the form of (a-b)(a2+ab+b2)=a3-b3

∴Here a=x13;b=1x13

(x1/3)3(1x1/3)3

x3/31x3/3

x1x


Question 18

(i) (xmxn)l(xnxl)m(xlxm)n

⇒(xm-n)l.(xn-l)2.(xl-n)n

⇒xml-nl.xmn-nl.xnl-nm

⇒xml-nl+mn-ml+nl-nm

⇒x0

⇒1


(ii) (xa+bxc)ab(xb+cxa)bc(xc+axb)ca

x(a+b)(ab)xc(ab)x(b+c)(bc)xa(bc)x(c+a)(ca)xb(ca)

xa2b2xacbcxb2c2xabacxc2a2xbcab

xa2b2+b2c2+c2a2xacbc+abac+bcab

x0x0

⇒1


Question 19

(i) lmxlxm×mnxmxn.nlxnxl

1mx1mmnxmnnlxnt

(xlm)1lm(xmn)1nm(xnl)1nl.

xlmlmxmnnmxnlnl

xlmlm+mnnm+nlnl

xn(lm)+l(mn)+m(nl)lmn

xnlnm+lmln+mnlnlmn

⇒x0

⇒1


(ii) (xaxb)a2+ab+b2(xbxc)b2+bc+c2(xcxa)c2+ac+a2

x(ab)(a2+ab+b2).x(bc)(b2+bc+c2).x(ca)(c2+ac+a2)

xa3b3xb3c3xc3a3

xa3b3+b3c3+c3a3

⇒x0

⇒1


(iii) (xaxb)a2ab+b2(xbxc)b2bc+c2(xcxa)c2ac+a2

x(acb))(a2ab+b2).x(bcc))(b2bc+c2).x(c(a))(c2ac+a2)

x(a+b)(a2ab+b2).x(b+c)(b2cb+c2).x(c+a)(c2ac+a2)

xa3+b3xb3+c3xc3+a3

xa3+b3+b3+c3+c3+a3

x2a3+2b3+2c3

x2(a3+b3+c3)


Question 20

(i) (a-1+b-1)÷(a-2-b-2)

(1a+1b)÷(1a21b2)

(b+aab)÷(b2a2a2b2)

(b+a)ab(b2a2)a2b2

(b+a)ab×(ab)2(b2a2)

b+aab(ab)2(b+a)(ba)

abba


(ii) 11+amn+1anm+1

11+amn+11+a(mn)

11+amn+11+1amn

11+amn+1amn+1amn

11+amn+amnamn+1

1+amn1+amn

⇒1


Question 21

(i) (a+b)1(a1+b1)=1ab

L.H.S⇒(a+b)-1(a-1+b-1)

1a+b(1a+1b)

1a+b(b+aab)

1a+b(a+bab)

1ab

⇒R.H.S


(ii) x+y+zx1y1+y1z1+z1x1=xyz

Sol :

L.H.S⇒x+y+zx1y1+y1z1+z1x1

x+y+z1xy+1yz+1xz

x+y+zz+x+yxyz

x+y+z(x+y+z)xyz

⇒xyz

⇒R.H.S


Question 22

Sol :

Given :

a=cz ; b=ax ; c=by

⇒a=cz

⇒a=(by)z  (∵c=by)

⇒a=(ax)yz

⇒a1=axyz

∴Bases are equal ; so exponents are also equal

∴xyz=1

Hence proved


Question 23

Sol :

Given : 

a=xyp-1 ; b=xyq-1 ; c=xyr-1

L.H.S⇒aq-r.br-q .cp-q

⇒(xyp-1)q-r.(xy)q-1(r-p).(xyr-1)p-q

⇒xypq-pr-q+r.xyqr-qp-r+p.xyrp-rq-p+q

⇒xypq-pr-q+r+qr-qp-r+p+rp-rq-p+q

⇒xy0

⇒1

⇒R.H.S

∴Hence proved


Question 24

Sol :

Given : 2x=3y=6-z

Let 2x=3y=6-z=k

2=k1/x

3=k1/y

6=k1/2

16=k1/2

12×3=k1/2

1k1x.k1y=k1/2

1=k1zk1x1y

k=k1x+1y+1z

1x+1y+12=0


Question 25

Sol :

Given : 2x=3y=6z

Let 2x=3y=6z=k

2=k1/x

3=k1/y

12=k1/z

223=k1/z

(k1/x)2k1/y=k1/z

k2xk1y=k1z

2x+1y=1z

2y+xxy=1z

2x=1z1y

2x=yzyz

x=2yzyz

∴Hence proved


Question 26

(i) (3x2)0

Sol :

⇒1


(ii) (xy)-2

1(xy)2

1x2y2


(iii) (-27aa)2/3

⇒-(33aa)2/3

⇒-(3a3)3 × 2/3

⇒-(3a3)2

⇒-32a3×2

⇒9a6


Question 27

Given : a=3 ; b=-2

(i) aa+bb

⇒33+(-2)-2

33+1(2)2

27+14

108+14

1094


(ii) ab+ba

⇒3-2+(-2)3

1328

198

1729

719


Question 28

Given : x=103×0.0099 ;

y=10-2×110

xy103×0.0099102×110

=103+2×0.0099110

105×0.0099110

990110

⇒√3

⇒3


Question 29

Given : x=9 ; y=2 ; z=8

x12y1z23

91221823

(32)12(12)(23)23

31222

3124

⇒6


Question 30

Given : x4y2z3=49392

2493922246962123482617433087310297343749771

⇒x4y2z3=243273

∵x,y,z are different primes

∴x=2 ; y=3 ; z=7


Question 31

Given : 3a6b4=axb2y

(a6b4)13=axb2y

a63b43=axb2y

x=63

⇒x=2


2y=43

y=43×2

y=23


Question 32

Given : 

⇒(p+q)-1(p-1+q-1)=paqb

1p+q(1p+1q)=paqb

1p+q(q+pqp)=paqb

1qp=paqb

⇒(qp)-1=pa.qb

⇒p-1.q-1=pa.qb

a=-1

b=-1

∴L.H.S⇒a+b+z

⇒-1+2-1

⇒0

=R.H.S


Question 33

Sol :

Given : 

(p1q2p2q4)7÷(p3q5p2q3)5=pxqy

(p7q2×7p2×7q4×7)÷(p3×5q5x5p2×5q3×5)=pxqy

(p7q14p14q28)÷(p15q25p10q15)=pxqy

⇒(p-7-14q14+28)÷(p15-10.q25+15)=pxqy

⇒(p-21.q42)÷(p5.q40)=px.qy

(p21q42p5q40)=pxqy

⇒(p-21-5.q42-20)=px.qy

⇒(p-26.q2)=px.qy

∴x=-26 ; y=2

∴x+y=-26+2

=-24


Question 34

(i) 52x+3=1

⇒52x+3=50  (∵50=1)

∴2x+3=0

⇒2x=-3

x=32


(ii) (13)x=44346

(13)x=256816

(13)x=169

(13)x=(13)2

(13)x=132

x=2

x=212


(iii) (35)x+1=12527

(35)x+12=5333

(35)x+12=(53)3

(35)x+12=(35)3

x+12=3

⇒x+1=-6

⇒x=-6-1

⇒x=-7


(iv) (34)2x+12=132

[(22)13]4x+12=132

(223)4x+12=125

24x+13=25

4x+13=5

⇒4x+1=-15

⇒4x=-15-1

⇒4x=-16

x=164

⇒x=-4


Question 35

(i) pq=(qp)12x

(pq)12=(pq)(12x)

(pq)12=(pq)1+2x

12=1+2x

1+2x=12

2x=12+1

2x=1+22

2x=32

x=32×2

x=34


(ii) 4x1×(0.5)32x=(18)x

22(x1)×(12)32x=(123)x

22x2×1232x=123x

22x2×22x3=23x

∴2x-2+2x-3=-3x

⇒4x-5=-3x

⇒4x+3x=5

⇒7x=5

x=57


Question 36

Given : 53x=125

⇒104=0.001

⇒53x=125

⇒53x=53

⇒3x=3

x=33

⇒x=1

∵10y=0.001

10y=(11000)

10y=(1103)

⇒10y=10-3

⇒y=-3

∴x=1 , y=-3


Question 37

Given : 9n323n27n33m23=127

32n323n33n33m23=133

32n+2+n33n33m8=133

33n+233n33m8=133

33n(321)33m8=133

33n(91)33m8=133

33n833m8=133

⇒33n.33=33m

⇒33(n+1)=33m
⇒m=n+1


Question 38

Given : 34x=(81)-1

⇒34x=(34)-1

⇒34x=3-4

⇒4x=-4

x=44

⇒x=-1

and 

101y=0.0001

101y=110000

101y=1104

101y=104

1y=4

y=14

⇒2-x.(16)y
⇒2-(-1)(24)-1/4
⇒21.2-1
212
⇒1

Question 39

Given : 3x+1=9x-2

⇒3x+1=32(x-2)

⇒3x+1=32x-4

⇒x+1=2x-4

⇒2x-x=1+4

⇒x=5


⇒21+x=21+5

=26

=64


Question 40

(i) 3(2x+1)-2x+2+5=0

⇒3.2x+3-2x.22+5=0

⇒3.2x+3-2x.4+5=0

⇒3.2x-4.2x+8=0

⇒-2x+8=0

⇒2x=8

⇒2x=23

⇒x=3


(ii) 3x=9.3y

⇒3x=32.3y

⇒3x=32+y

⇒x=2+y

and 8.2y=4x

⇒232y=22x

⇒3+y=2x

x=3+y2

Comments

Popular posts from this blog

ML Aggarwal Solution Class 10 Chapter 15 Circles Exercise 15.1

ML Aggarwal Solution Class 9 Chapter 20 Statistics Exercise 20.2

ML Aggarwal Solution Class 9 Chapter 3 Expansions Exercise 3.2