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02. Square Root and Cube Roots

Jun 03, 2018

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  • 8/12/2019 02. Square Root and Cube Roots

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    :KF@Q^HZ

    [PX ZH ZMM^ KXJH ZMM^[

    Gipmrt`bt E`kts @bn Emriuc`h ? [qu`rh Zmmt ? Ge x: 9 y, Th s`y tf`t tfh squ`rh

    rmmt me y gs x `bn wh wrgth, xy . ^fus, > 9:,

    79;, 07590>.

    Kujh Zmmt ? ^fh kujh rmmt me ` dgvhb buijhr xgs tfh buijhr wfmsh kujh gs x. Th nhbmth tfh kujh

    rmmt me x jy ;

    x .^fus, ; 69 ; ::: 9:, ; 444 94 htk.

    Bmth ?

    0. xy 9 x y

    :.y

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    HV@IQCH[Hx.0 Hv`cu`th 516> jy e`ktmrgz`tgmb ihtfmn,

    [mc. Ihtfmn ? Hxprhss tfh dgvhb buijhr `s tfhprmnukt me prgih e`ktmrs. Bmw, t`lh tfh prmnukt

    me tfhsh prgih e`ktmrs kfmmsgbd mbh mut me hvhry

    p`gr me tfh s`ih prgihs. ^fgs prmnukt dgvhs tfh

    squh`rh rmmt me tfh dgvhb buijhr.

    ^fus, rhsmcvgbd 516> gbtm prgih e`ktmrs, wh dht?

    516>

    ;1>:

    08:0

    814

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    516> 9 (: ; 0 ; ) 9 4 6 .

    Hx.: Egbn tfh squ`rh rmmt me 0>40;57.

    [mc. Hxpc b`tgmb ?Gb tfh dgvhb buijhr, i`rl mee tfh

    ngdgts gb p`grs st`rtgbd ermi tfh ubgts ngdgt. H`kf

    p`gr `bn tfh rhi`gbgbd mbh ngdgt gs k`cchn ` phrgmn.

    Bmw, 0: 9 0. Mbh sujtr`ktgbd, wh dht 1 `s

    rhi`gbnhr. Bmw, jrgbd nmwb tfh bhxt phrgmn g.h., >4.

    Bmw, trg`c ngvgsmr gs 0 : 9 : `bn trg`cc ngvgnhbn

    gs >4.

    [m, wh t`lh :: `s nvsmr `bn put : `s qumtghbt.

    ^fh rhi`gbnhr gs ;.

    Bhxt, wh jrgbd nmwb tfh bhxt phrgmn wfgkf gs 0;.

    Bmw, trg`c ngvgsmr gs 0: : 9 :> `bn trg`c ngvgnhbn

    gs ;0;. [m, wh t`lh :>0 `s ngvgnhbn `bn 0 `s

    qumtghbt.

    ^fh rhi`gbnhr gs 4:.

    Jrgbd nmwb tfh bhxt phrgmn g.h., 57.

    Bmw, tfh trg`c ngvgsmr gs 0:0 : 9 :>: `bn tfh trg`c

    ngvgnhbn gs 4:57. [m, wh t`lh ; `s qu`tghbt `bn

    :>:; `s ngvgnsmr. ^fh rhi`gbnhr gs tfhb zhrm.

    Fhbkh, 0>40;57 9 0:0;.

    0>40;57 (0:0;

    0>4>>

    ;0;:>0

    4:574:57

    0

    ::

    :>0

    :>:;

    Hx.; Hv`cu`th ? 05780:>6 .

    [mc. Dgvhb hxprhssgmb

    9 0;80:>6 9 5>:>6 9 6:>6 9

    :85905.

    Hx.> Ge `*j*k 9

    0k

    ;j:`

    , tfhb egbn tfh v`cuh

    me 5*08*;.

    [mc. 5*08*; 9

    0;

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    9

    >

    0669

    >

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    9>

    0: 9 ;.

    Hx.8 Egbn tfh v`cuh me05

    70 .

    [mc.05

    70 9

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    8

    05

    :8

    05

    :8 .

    Hx.5 Tf`t gs tfh squ`rh rmmt me 1.1117=

    [mc. 1117.1 9 01111

    79

    01111

    79

    011

    ;9 1.1;

  • 8/12/2019 02. Square Root and Cube Roots

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    Hx.4 Hv`cu`th :745.048 .

    [mc. Ihtfmn ?Th i`lh hvhb buijhr me nhkgi`c pc`khs

    jy `eegxgbd ` zhrm, Ge bhkhss`ry. Bmw wh i`rl mee

    phrgmns `bn hxtr`kt tfh squ`rh rmmt `s sfmwb.

    048.:745 (0;.:>

    04857

    5:78:>

    0184501845

    0

    :;

    :5:

    :5>>

    :745.048 9 0;.:>.

    Hx.6 Tf`t wgcc kmih gb pc`kh me quhstgmb i`rl gb h`kf

    me tfh emccmwgbd quhstgmbs=

    (g) :=

    >.;: (gg) ;.0:(=)8>7.65 :

    [mc. (g) Chtx

    >.;:9 :. ^fhb,

    x

    >.;:9 > >x

    9 ;:.> x 9 6.0

    (gg) Cht :x8>7.65 90:.;.

    ^fhb, 7.; + :x8 90:.; :x8

    9 0:.; - 7.; 9 ;

    8+x: 9 7 x: 9 7 - 8 9 > x9 > 9:.

    Hx.7 Egbn tfh v`cuh me110:0.1

    :67.1.

    [mc.110:0.1

    :67.19

    110:0.1

    :6711.19

    0:0

    :67119

    00

    041

    Hx.01 Ge0>>

    x0 9

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    0;, tfhb egbn tfh v`cuh me x.

    [mc.0:

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    x x 9 :8.

    Hx.00 Egbn tfh v`cuh me ; uptm tfrhh pc`khs me

    nhkgi`c.

    [mc.

    ;.111111 (0.4;:0:11067

    001101:7

    401157:>

    0

    :4

    ;>;

    ;>5:

    ; 9 0.4;:.

    Hx.0: Ge ;90.4;:, egbn tfh v`cuh me

    48>6:

    007: k m rrh kt t m ; p c `k h s m e

    nhkgi`c.

    [mc. 48>6:

    007:

    ;:8;05:

    0;5>

    ;8;>:

    0;6

    4;:.0;;:;;

    Hx.0; Hv`cu`th ?1:0.17.01104.

    7.061168.8.7

    .

    [mc. Dgvhb hxp. 91:0.17.01104.

    711.061168.8.7

    Bmw, sgbkh tfh sui me nhkgi`c pc`khs gb tfh

    buihr`tmr `bn nhbmigb`tmr `bn ubnhr tfh r`ng`c

    sgdb gs tfh s`ih, wh rhimvh tfh nhkgi`c.

    Dgvhb hxp. 9:00704

    067116878

    9 71188

    9 8 ;1 9 081.

    Hx.0> [gipcgey ?

    )Y78.07)(:8.1():8.1\(Y)0.6()0.0:\( :::

    [mc. Dgvhb hxp.

    9 :.:1:8.1

    >:.:1

    78.07:8.1:8.1

    )0.60.0:)(0.60.0:(

    9 >05:8

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    Hx.08 Ge x 9 0+ : `bn y 9 0- : , egbn tfh v`cuh me

    (x:+y:).

    [mc. x: + y: 9 (0 + : ): 9:(0): + :):( Y 9 : ; 95.

    Hx.05 Hv`cu`th 7.1 uptm tfrhh pc kh me nhkgi`c.[mc.

    1.711111 (.7>660

    7114;505>110801>

    7

    06>

    0666

    7.1 91.7>6

    Hx.04 Ge 08

    9 ;.66, egbn tfh v`cuh me;

    8.

    [mc.;

    89

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    66.;

    ;

    08

    ;;

    ;8

    90.:7;; .......... 9 ;:7.0 .

    Hx.06 Egbn tfh ch`st squ`rh buijhr wfgkf gs hx`ktcy

    ngvgsgjch jy 01, 0:, 08 `bn 06.

    [mc. C.K.I. me 01, 0:, 08, 06 9 061. Bmw,

    061 9 : : ; ; 8 9 :: ;: 8.

    ^m i`lh gt ` phrehkt squ`rh, Gt iust jh iuctgpcghn

    jy 8.

    Zhqugrhn buijhr 9 (:: ;: 8:)9 711.

    Hx.07 Egbn tfh drh`thst buijhr me egvh ngdgts wfgkf gs

    ` phrehkt squ`rh.

    [mc. Drh thst buijhr me 8 ngdgts gs 77777.

    777505677;485

    0>;

    ;

    5:5

    77777 (;05

    50

    Zhqugrhn buijhr 9 (77777 0>;) 9 77685.

    Hx.:1 Egbn tfh si`cchst buijhr tf`t iust jh `nnhn tm

    0461 tm i`lh gt ` phrehkt squ`rh.

    [mc.

    05

    061

    05>05

    >

    6:

    0461 (>:

    Zhqugrhn tm jh `nnhn 9 (>;): 04619 06>7 0461 9 57.

    Hx.:0 Ge : 9 0.>0>:, egbn tfh v`cuh me )::(

    :

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    9

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    9 )0:(:

    )0:(:

    9 (0.>0>:0) 9 1.>0>:.

    Hx.:: Ge x 9

    ;8

    ;8`bn y 9

    ;8

    ;8, egbn tfh

    v`cuh me (x: + y:).

    [mc. x 9);8(

    );8(

    );8(

    );8(

    );8(

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    9 .08>:

    08:;8

    y 9);8(

    );8(

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    9 .08>:

    08:;8

    x: + y: 9 (> + :)08( 9: Y)08()>\( ::

    9 : ;0 9 5:.

    Hx.:; Egbn tfh kujh me rmmt me :4>>.

    [mc. Ihtfmn ? Zhsmcvh tfh dgvhb buijhr `s tfh

    prmnukt me prgih e`ktmrs `bn t`lh tfh prmnukt me

    prgih e`ktmr. Zhsmcvgbd :4>> `s tfh prmnukt me

    prgih e`ktmrs, wh dht ?

    ; :4>> 9 :; 4;.

    :4>>0;:4

    565;>;

    >74

    :

    :

    :44

    ; :4>> 9 : 4 9 0>.HVHZKGKH

    Hx.:> Jy wf`t ch st buijhr >;:1 jh iuctgpcghn tm mjt gb

    ` buijhr wfgkf gs ` phrehkt kujh =

    [mc. Kch`rcy, >;:1 9 :; ;; :: 8.

    ^m i`lh gt ` phrehkt kujh, gt iust jh iuctgpcghn

    jy : 8: g.h., 81.

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    0. =8;6:>

    (@) :1: (J) :;: (K) :>: (N) ;;:

    :. ^fh squ`rh rmmt me 5>117 gs -

    (@) :8; (J) ;>4 (K) ;5; (N) 61;

    ;. ^fh v`cuh me ::808>016:801

    gs -

    (@) > (J) 5 (K) 6 (N) 01

    >. Hv`cu`th ? 707:0>0 -

    (@) ; (J) 8 (K) 5 (N) 5.7

    8. :>10045 gs hqu`c tm -

    (@) 0> (J) 08 (K) 06 (N) :>

    5.

    075

    00

    :8

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    5:8gs hqu`c tm -

    (@) 8 (J) 5 (K) 6 (N) 00

    4. =60

    05

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    ::8

    (@)>6

    0(J)

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    8(K)

    05

    8(N) Bmbh

    6. ^fh squ`rh rmmt me (:4:: 0:6:) gs -

    (@) 0>> (J) :11 (K) :>1 (N) :85

    7. Ge xyyxy*x , tfh v`cuh me :>*5 gs -

    (@) >0 (J) >: (K) >; (N) >>

    01. Ge y 9 8, tfhb wf`t gs tfh v`cuh me 01y :; yy =

    (@) 81 : (J) 011 (K) :11 8 (N) 811

    00.>

    0001 9=

    (@) 01.:8 (J) 01.8 (K) 00.08 (N) 07.8

    0:.7

    0

    60

    :89 =

    (@);

    :(J)

    7

    >(K)

    60

    05(N)

    60

    :8

    0;. ^fh ngdgt gb tfh ubgts pc`kh gb tfh squ`rh rmmt me

    08645 gs -

    (@) : (J) > (K) ; (N) Bmbh

    0>. Fmw i`by twm ngdgt-buijhrs s`tgsey tfgs

    prmphrty ? ^fh c`st ngdgt (ubgts ngdgt) me tfh

    squ`rh me tfh twm-ngdgt buijhr gs 6 =

    (@) 0 (J) : (K) ; (N) Bmbh

    08. Tf`t gs tfh squ`rh rmmt me 1.05 =

    (@) 1.11> (J) 1.1> (K) 1.> (N) >

    05. ^fh v`cuh me 111>450.1 gs -

    (@) 1.111:0 (J) 1.11:0 (K) 1.1:0 (N) 1.:0

    04. 1111>450.1 hqu cs -

    (@) 1.11157 (J) 1.1157 (K) 1.1517 (N) 1.157

    06. 0.8: 1::8.1 9 =

    (@) 1.1;48 (J) 1.;;48 (K) ;.:48 (N) ;:.48

    07. 115>.110.1 9 =

    (@) 1.1; (J) 1.; (K) 1.>: (N) Bmbh

    :1. ^fh v`cuh me 10.1

    + 60.1

    + :0.0

    + 1117.1

    gs -

    (@) :.1; (J) :.0 (K) :.00 (N) :.0;

    :0. 1110.:8.:11:8. 9 =

    (@) .111148 (J) .1148 (K) .148 (N) Bmbh

    ::. 85:8.0 9 =

    (@) 0.18 (J) 0.:8 (K) 0.>8 (N) 0.88

    :;. Ge 11111545. 9 . 11 : 5, t f h sq u` rh rm mt m e

    54,51,111 gs -

    (@):5

    0(J) :5 (K) :51 (N) :511

    :>. Ge 06::890;8, tfhb tfh v`cuh me

    11106::8.1106::8.16::8.0:8.06:(

    gs ?

    (@) 0.>77768 (J) 0>.7768

    (K) 0>7.768 (N) 0>77.68

    HVHZKG[H

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    :8. Dgvhb tf`t 0;9;.518 `bn 0;1 900.>1, egbn tfh

    v`cuh me 10;.10;11;.0 -

    (@) ;5.05> (J) ;5.;1> (K) ;4.05> (N) ;4.;1>

    :5. Gex

    8: 9:67057 , tfh v`cuh me x gs -

    (@) 8: (J) ;6 (K) 5: (N) 56

    :4. Emr wf`t v`cuh me * tfh st`thihbt

    0;8*

    08*

    90

    gs truh =

    (@) 08 (J) :8 (K) ;8 (N) >8

    :6. Tfgkf buijhr k`b rhpc`kh jmtf tfh quhstgmb

    i`rls gb tfh hqu`tgmb;:=

    =:

    0>

    .

    (@) 0 (J) 4 (K):

    04 (N) Bmbh

    :7. Tf`t sfmucn kmih gb pc kh me jmtf tfh quhstgmb

    i`rls gb tfh hqu`tgmb=

    05:

    0:6

    = .

    (@) 0: (J) 0> (K) 0>> (N) 075

    ;1. Ge 1.0; p: 9 0;, tfhb p hqu`cs ?

    (@) 1.10 (J) 1.0 (K) 01 (N) 011

    ;0. Tf t buijhr sfmucn jh ngvgnh jy :8.1 tm

    dgvh tfh rhsuct `s :8 =

    (@) 0:.8 (J) :8 (K) 81 (N) 0:8

    ;:. Ge b; 94:7, tfhb tfh v`cuh me b gs ?

    (@) 5 (J) 6 (K) 01 (N) 0:

    ;;. Ge x0>06 96>, tfhb x hqu cs ?

    (@) :: (J) :> (K) :6 (N) ;:

    ;>. :6 = + 0>:5 9>

    ;me :64:.

    (@) 845 (J) 545 (K) 0;75 (N) 0>>>

    ;8.;7

    8>

    057

    =

    (@) 016 (J) ;:> (K) :705 (N) >611

    ;5. Ge >>0x 9 1.1:, tfhb tfh v`cuh me x gs -

    (@) 1.045> (J) 0.45> (K) 0.5> (N) :.5>

    ;4. :.1

    =

    075.1

    (@) 1.>7 (J) 1.4 (K) >.7 (N) Bmbh

    ;6. ;.0=1.1057

    (@) 01 (J) 011 (K) 0111 (N) Bmbh

    ;7. Ge x1508.0;57 9;4.:8, tfhb x gs hqu`c tm-

    (@) 01-0 (J) 01-: (K) 01-; (N) Bmbh

    >1. Ge ,4):y)(0x( x `bn y jhgbd pmsgtgvh

    wfmch buijhrs, tfhb tfh v`cuhs me x `bn y

    rhsphktgvhcy `rh -

    (@) 6, 8 (J) 08, 0: (K) ::, 07 (N) Bmbh

    >0. Ge `>.1>. 9 .11> .> j , tfhb j

    `gs -

    (@) 0501-; (J) 0501->

    (K) 0501-8 (N) Bmbh

    >:. ^frhh-egetf me tfh squ`rh me ` khrt gb buijhr gs

    0:5.08. Tf`t gs tfh buijhr =

    (@) 0>.8 (J) 48.57 (K) 0>8 (N) :01.:8

    >;. =11057.1

    ;50.1

    (@)0;

    7.0(J)

    0;

    07(K)

    0;1

    7.0(N)

    0;

    071

    >>.:67.1

    >.>6gs hqu`c tm -

    (@)04

    80 (J)

    04

    00:

    (K)04050: (N)

    0440:7

    .>8. Ge0;

    0>

    057

    x0 , tfhb x gs hqu`c tm -

    (@) 0 (J) 0;

    (K) :4 (N) Bmbh me tfhsh

    >5. Ge:4

    x0

    4:7

    880 , tfhb tfh v`cuh me x gs -

    (@) 0 (J) ; (K) 8 (N) 4

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    >4. ^fh v`cuh me : uptm tfrhh pc`kh me nhkgi`c gs-

    (@) 0.>01 (J) 0.>0: (K) 0.>0; (N) 0.>0>

    >6. )0:48:4:( gs hqu c tm -

    (@) ; (J) ;: (K) ;; (N) ;>

    >7. Jy fmw iukf nmhs 060: hxkhhn :; =

    (@) ;>: (J) ::;

    (K) ):;(: (N) ):;(;

    81. =75

    :05:>

    (@) 5: (J) : (K) :5 (N) 5

    :

    80. ^fh v`cuh me5;>8

    00:61 gs -

    (@)>

    ;(J)

    ;

    00 (K)

    7

    40 (N)

    >

    ;0

    8:. Ge 8; + 0:8 904.66, tfhb wf`t wgcc jh tfh v`cuh

    me 8561 =

    (@) 0;.>0 (J) :1.>5 (K) :0.55 (N) ::.;5

    8;. 7681 gs hqu`c tm -

    (@) 5;.48 (J) 58.78 (K) 41 (N) 41.:8

    8>. Dgvhb : 90.>0>. ^fh v`cuh me

    ;::6 0:6; 81> gs -

    (@) 6.:>5 (J) 6.>6> (K) 6.8:5 (N) 6.645

    88. ^fh `pprmxgi`th v`cuh me76

    :0:

    :6:

    :0; gs -

    (@) 0.1518 (J) 0.14:4 (K) 0.5114 (N) 0.51:5

    85.:8.5115>.

    >6>.160.

    gs hqu`c tm -

    (@) 1.7 (J) 1.77 (K) 7 (N) 77

    84.;1>14.1

    >::1>.1

    gs hqu`c tm -

    (@)5

    0(J) 1.15 (K) 1.5 (N) 5

    86.5>7.4:1:67.185:8.0

    5:>.>;:>.1160.1

    gs hqu`c tm -

    (@) 1.1:> (J) 1.:> (K) :.> (N) :>

    87.07.1104.

    168.8.7

    hqu`cs -

    (@) .18 (J) 8 (K) 81 (N) 811

    51. ^fh v`cuh me :::

    :::

    )115>.1()1:0.1()11;.1(

    )158.1():0.1()1;.1(

    gs -

    (@) 1.0 (J) 01 (K) 01: (N) 01;

    50. ^fh squ`rh rmmt me (4+ 8; )(4- 8; ) gs -

    (@) 8 (J) : (K) > (N) 8;

    5:.

    :

    ;

    0;

    sgipcgeghs tm -

    (@)>

    ;(J)

    ;

    >

    (K);

    >(N) Bmbh me tfhsh

    5;.

    :

    :

    0

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    gs hqu`c tm -

    (@):

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    :

    08

    5>. Ge ` 9 1.01;7, tfhb tfh v`cuh me `;0`>`> :

    gs -

    (@) 1.01;7 (J) 1.:146 (K) 0.01;7 (N):.01;7

    58. ^fh squ`rh rmmt me 48.10

    )48.1( ;

    \1.48+(1.48):+0Y gs-

    (@) 0 (J) : (K) ; (N) >

    55. Ge ;` 9 >j 9 5` `bn ` + j + k 9 :7:4 , tfhb

    ::: kj` gs -

    (@) :7; (J) 60

    (K) 64 (N) Bmbh me tfhsh

    54. ^fh squ`rh rmmt me >.1 gs -

    (@) 5.1 (J) 4.1 (K) 6.1 (N) 7.1

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    56. Tfgkf mbh me tfh emccmwgbd buijhrs f`s r`tgmb`c

    squ`rh rmmt =

    (@) 1.> (J) 1.17 (K) 1.7 (N) 1.1:8

    57. ^fh v`cuh me >.1 gs -

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