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Operation of Indices

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HỦY DIỆT THE WORLD
09/02/2018 at 21:26
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solution of the equation : \(\dfrac{x+2}{89}+\dfrac{x+5}{86}>\dfrac{x+8}{83}+\dfrac{x+11}{80}\)

Operation of Indices

  • ...
    FA KAKALOTS 09/02/2018 at 21:34

    We have :

    \(=\dfrac{x+2}{89}+1+\dfrac{x+5}{86}+1>\dfrac{x+8}{83}+1+\dfrac{x+11}{80}+1\)

    \(=\dfrac{x+91}{89}+\dfrac{x+91}{86}>\dfrac{x+91}{83}+\dfrac{x+91}{80}\)

    \(=\dfrac{x+91}{89}+\dfrac{x+91}{86}-\dfrac{x+91}{83}-\dfrac{x+91}{80}>0\)

    \(=\left(x+91\right)\left(\dfrac{1}{89}+\dfrac{1}{86}-\dfrac{1}{83}-\dfrac{1}{80}\right)>0\)

    To \(\dfrac{1}{89}+\dfrac{1}{86}-\dfrac{1}{83}-\dfrac{1}{80}\) different 0


    \(=x+91>0\)

    \(=x< -91\)

    HỦY DIỆT THE WORLD selected this answer.
  • ...
    Fc Alan Walker 10/02/2018 at 14:16

    e have :

    =x+289+1+x+586+1>x+883+1+x+1180+1

    =x+9189+x+9186>x+9183+x+9180

    =x+9189+x+9186−x+9183−x+9180>0

    =(x+91)(189+186−183−180)>0

    To 189+186−183−180

    different 0


    =x+91>0

    =x<−91


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Neymar
19/03/2017 at 11:58
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Given m2 + m = \(\dfrac{1}{4}\), find the value of 8m4 + 20m3 + 13m2.

Operation of Indices


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Neymar
19/03/2017 at 11:57
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Suppose a + b = 5 and a3 + b3 = 35, find the value of a2 + b2.

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:28

    a + b = 5

    <=> (a+b)3=125

     <=> a3 + 3a2b + 3ab2 + b3 = 125

    <=> 3ab(a + b) + 35 = 125 (a3 + b3 = 35)

    <=> 15ab = 90 (a + b = 5)

    <=> ab = 6

    a + b = 5

    <=> (a + b)2 = 25

    <=> a2 + 2ab + b2 = 25

    <=> a2 + 12 + b2 = 25

    <=> a2 + b2 = 13

  • ...
    Hồ Thu Giang 19/03/2017 at 14:26

    a + b = 5

    <=> \(\left(a+b\right)^3=125\) <=> a3 + 3a2b + 3ab2 + b3 = 125

    <=> 3ab(a + b) + 35 = 125 (a3 + b3 = 35)

    <=> 15ab = 90 (a + b = 5)

    <=> ab = 6

    a + b = 5

    <=> (a + b)2 = 25

    <=> a2 + 2ab + b2 = 25

    <=> a2 + 12 + b2 = 25

    <=> a2 + b2 = 13


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Neymar
19/03/2017 at 11:56
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If 3x + 5y + 3 = 0 find the value of 83x+2 . 32y

Operation of Indices


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Neymar
19/03/2017 at 11:54
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If ax + a-x = 3, find the value of a2x + a-2x.

Operation of Indices


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Lionel Messi
19/03/2017 at 11:52
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Suppose (x+y)2 = 18 and (x-y)2 = 12. Find the value of x2 + y2.

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:29

    We have : (x + y)2 - (x - y)2 = 18 - 12 

    (x2 + 2xy + y2) - (x2 - 2xy + y2) = 6

                                           4xy       = 6

                                           2xy       = 3

    => x2 + y2 = 18 - 3 = 15

                         

  • ...
    Phan Thanh Tinh Coordinator 23/03/2017 at 17:45

    We have : (x + y)2 - (x - y)2 = 18 - 12 

    (x2 + 2xy + y2) - (x2 - 2xy + y2) = 6

                                           4xy       = 6

                                           2xy       = 3

    => x2 + y2 = 18 - 3 = 15

                          


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Lionel Messi
19/03/2017 at 11:51
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It is given 3x = \(\dfrac{2}{9}\) and 3y = 27. Find the value of 33x+2y.

Operation of Indices

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    FA KAKALOTS 06/02/2018 at 12:29

    33x+2y=33x×32y=(3x)3×(3y)2=(29)3×272

    =23×3636=8

  • ...
    Phan Thanh Tinh Coordinator 23/05/2017 at 09:25

    \(3^{3x+2y}=3^{3x}\times3^{2y}=\left(3^x\right)^3\times\left(3^y\right)^2=\left(\dfrac{2}{9}\right)^3\times27^2\)

    \(=\dfrac{2^3\times3^6}{3^6}=8\)


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Lionel Messi
19/03/2017 at 11:50
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Find the value of a in 10x3 + ax2 + 5x + 2 by expanding (2x2 + 1)(5x + 2)

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:29

    We have : (2x2 + 1)(5x + 2) = 10x3 + 4x2 + 5x + 2

    So a = 4

  • ...
    Phan Thanh Tinh Coordinator 22/03/2017 at 22:18

    We have : (2x2 + 1)(5x + 2) = 10x3 + 4x2 + 5x + 2

    So a = 4


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Lionel Messi
19/03/2017 at 11:49
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Solve for the value of m in each of the following:

(a) 5m . 252m = 1252m-1

(b) 5m . 25m . 125m . 625m = 530 

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:29

    (a) 5m.252m = 1252m - 1

    => 5m.54m = (53)2m - 1

    => 55m = 56m - 3

    => 5m = 6m - 3

    => m = 3

    (b) 5m.25m.125m.625m = 530

    => (5.25.125.625)m = 530

    => (5.52.53.54)m = 530

    => 510m = 530

    => 10m = 30

    => m = 3

  • ...
    Phan Thanh Tinh Coordinator 22/03/2017 at 22:22

    (a) 5m.252m = 1252m - 1

    => 5m.54m = (53)2m - 1

    => 55m = 56m - 3

    => 5m = 6m - 3

    => m = 3

    (b) 5m.25m.125m.625m = 530

    => (5.25.125.625)m = 530

    => (5.52.53.54)m = 530

    => 510m = 530

    => 10m = 30

    => m = 3


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Lionel Messi
19/03/2017 at 11:47
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If a = 5x2 - 4x + 2 and b = 5x2 - 4x - 3, is a > b true?

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:30

    Because 5x2−4x=5x2−4x and 2>−3

     => a > b 

    You are right (True).

  • ...
    Lê Quốc Trần Anh Coordinator 06/09/2017 at 13:11

    Because \(5x^2-4x=5x^2-4x\) and \(2>-3\) => a > b 

    You are right (True).


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Lionel Messi
19/03/2017 at 11:45
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If 2a x 3b = 2c x 3d = 36, find the value of (a-1)(d+1)+(b+1)(c-1).

Operation of Indices


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Lionel Messi
19/03/2017 at 11:43
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If (x+y)2 = 9 and (x-y)2 = 1, x2 + y2 = ?

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:30

    (x + y)2 = 9

    <=> x2 + 2xy + y2 = 9 (1)

    (x - y)2 = 1

    <=> x2 - 2xy + y2 = 1 (2)

    (1) - (2) => 4xy = 8 => xy = 2

    => x2 + 2xy + y2 = 9

    <=> x2 + 4 + y2 = 9

    <=> x2 + y2 = 5

  • ...
    Hồ Thu Giang 19/03/2017 at 14:36

    (x + y)2 = 9

    <=> x2 + 2xy + y2 = 9 (1)

    (x - y)2 = 1

    <=> x2 - 2xy + y2 = 1 (2)

    (1) - (2) => 4xy = 8 => xy = 2

    => x2 + 2xy + y2 = 9

    <=> x2 + 4 + y2 = 9

    <=> x2 + y2 = 5


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Lionel Messi
19/03/2017 at 11:42
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what is the value of n in 32n+1+4n=259?

Operation of Indices


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Lionel Messi
19/03/2017 at 11:41
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What is the ones digit of 725?

Operation of Indices

  • ...
    FA KAKALOTS 06/02/2018 at 12:30

    We have : 725 = (74)6.7

    The units digit of 74 is 1 (74 = 2401),so the units digit of (74)6 is 1

    => The units digit of 725 is 7

  • ...
    Phan Thanh Tinh Coordinator 23/03/2017 at 17:47

    We have : 725 = (74)6.7

    The units digit of 74 is 1 (74 = 2401),so the units digit of (74)6 is 1

    => The units digit of 725 is 7


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steve jobs
18/03/2017 at 10:15
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if a+b = 4 and \(a^3+b^3=28,a^2+b^2=?\)

\([hint:\left(a+b\right)^2=a^2+2ab+b^2\)\(\left(a+b\right)^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

Operation of Indices

  • ...
    mathlove 18/03/2017 at 11:04

    By the assumption  \(\left\{{}\begin{matrix}a+b=4\\a^3+b^3=28\end{matrix}\right.\Rightarrow a^2-ab+b^2=\dfrac{28}{4}=7\Rightarrow2a^2-2ab+2b^2=14\).

    Other hand, \(a+b=4\Rightarrow a^2+2ab+b^2=16\). So:

               \(3\left(a^2+b^2\right)=\left(2a^2-2ab+2b^2\right)+\left(a^2+2ab+b^2\right)=14+16\Rightarrow a^2+b^2=10\)

    steve jobs selected this answer.
  • ...
    FA KAKALOTS 06/02/2018 at 12:30

    By the assumption  {a+b=4a3+b3=28⇒a2−ab+b2=284=7⇒2a2−2ab+2b2=14

    .

    Other hand, a+b=4⇒a2+2ab+b2=16

    . So:

               3(a2+b2)=(2a2−2ab+2b2)+(a2+2ab+b2)=14+16⇒a2+b2=10


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steve jobs
18/03/2017 at 10:12
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Given m+n = 11 and mn = 9, find the value of \(m^2+n^2\).

Operation of Indices

  • ...
    mathlove 18/03/2017 at 11:05

    \(m^2+n^2=\left(m+n\right)^2-2mn=103\)

    steve jobs selected this answer.
  • ...
    FA KAKALOTS 06/02/2018 at 12:31

    m2+n2

    = (m+n)2-2mn

    = 112-2.9

    =121-18

    =103

  • ...
    Nguyễn Trần Thành Đạt 18/03/2017 at 16:44

    m2+n2

    = (m+n)2-2mn

    = 112-2.9

    =121-18

    =103


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steve jobs
18/03/2017 at 10:10
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Given 3x + 5y = 2, find the value of \(27^x.243^y\).

Operation of Indices

  • ...
    mathlove 18/03/2017 at 10:37

    \(27^x.243^y=\left(3^3\right)^x\left(3^5\right)^y=3^{\left(3x+5y\right)}=3^2=9\)

    Selected by MathYouLike
  • ...
    FA KAKALOTS 06/02/2018 at 12:31

    27x.243y=(33)x(35)y=3(3x+5y)=32=9


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steve jobs
18/03/2017 at 10:09
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Given \(a^m=\dfrac{1}{3}\) and \(a^n=6\), find the value of \(2^{2m+3n}\).

Operation of Indices

  • ...
    mathlove 18/03/2017 at 10:43

    We have    \(a^m=\dfrac{1}{3},a^n=6\Rightarrow2=6.\dfrac{1}{3}=a^n.a^m=a^{m+n}\), so  

                                                         \(2^{2m+3n}=\left(a^{m+n}\right)^{2m+3n}=a^{\left(m+n\right)\left(2m+3n\right)}\) .

    Selected by MathYouLike
  • ...
    FA KAKALOTS 06/02/2018 at 12:31

    We have    am=13,an=6⇒2=6.13=an.am=am+n

    , so  

                                                         22m+3n=(am+n)2m+3n=a(m+n)(2m+3n)

     .


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steve jobs
18/03/2017 at 10:07
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Rank \(2^{777}\),\(3^{555}\)and \(4^{444}\) in ascendung order in terms of their values.

Operation of Indices

  • ...
    mathlove 18/03/2017 at 10:52

    Put   \(a=2^{777},b=3^{555},c=4^{444}\). We have :

    \(a=2^{777}=\left(2^7\right)^{111}=128^{111},b=3^{555}=\left(3^5\right)^{111}=243^{111},c=4^{444}=\left(4^4\right)^{111}=256^{111}\Rightarrow a< b< c\).

    So:  \(2^{777}< 3^{555}< 4^{444}\).

    Selected by MathYouLike
  • ...
    FA KAKALOTS 06/02/2018 at 12:31

    Put   a=2777,b=3555,c=4444

    . We have :

    a=2777=(27)111=128111,b=3555=(35)111=243111,c=4444=(44)111=256111⇒a<b<c

    .

    So:  2777<3555<4444

    .


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steve jobs
18/03/2017 at 10:04
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Solve the value of the unknown in each of the following:

(a) \(3^x=243\)

(b) \(5^m=625\)

(c) \(16^x.5^{2x}=400\)

(d) \(2^{2n+2}+2^{2n}=160\)

Operation of Indices

  • ...
    mathlove 18/03/2017 at 11:13

    a) \(3^x=243=3^5\Leftrightarrow x=5\).

    b) \(5^m=625\Leftrightarrow5^m=5^4\Leftrightarrow m=4\).

    c) By  \(16^x.5^{2x}=16^x.25^x=\left(16.25\right)^x=400^x\), so the done equation equivalant to   

                                                              \(400^x=400^1\Leftrightarrow x=1\).

    d) \(2^{2n+2}+2^{2n}=160\Leftrightarrow2^{2n}\left(2^2+1\right)=160\Leftrightarrow2^{2n}=2^5\Leftrightarrow2n=5\Leftrightarrow n=2,5.\)

    Selected by MathYouLike
  • ...
    FA KAKALOTS 06/02/2018 at 12:32

    a) 3x=243=35⇔x=5

    .

    b) 5m=625⇔5m=54⇔m=4

    .

    c) By  16x.52x=16x.25x=(16.25)x=400x

    , so the done equation equivalant to   

                                                              400x=4001⇔x=1

    .

    d) 22n+2+22n=160⇔22n(22+1)=160⇔22n=25⇔2n=5⇔n=2,5.

  • ...
    Hạo Nam 24/03/2017 at 17:56
    a) 3x=243=35⇔x=53x=243=35⇔x=5.

    b) 5m=625⇔5m=54⇔m=45m=625⇔5m=54⇔m=4.

    c) By  16x.52x=16x.25x=(16.25)x=400x16x.52x=16x.25x=(16.25)x=400x, so the done equation equivalant to   

                                                              400x=4001⇔x=1400x=4001⇔x=1.

    d) 22n+2+22n=160⇔22n(22+1)=160⇔22n=25⇔2n=5⇔n=2,5.22n+2+22n=160⇔22n(22+1)=160⇔22n=25⇔2n=5⇔n=2,5.

     Selected by MathYou


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