Evaluate $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$












0












$begingroup$


I need to find the value of



$$( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



I tried using the identity $a^{3} - b^{3}$ but couldn't reach very far and got stuck.






radicals







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$endgroup$












  • $begingroup$
    What equation? What is there to solve?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:11










  • $begingroup$
    I mean to find its value.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:12










  • $begingroup$
    Edited the question. My bad.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:13










  • $begingroup$
    I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:14












  • $begingroup$
    Is there a specific form you are looking for in the final answer?
    $endgroup$
    – Aditya Dua
    Dec 12 '18 at 17:22
















0












$begingroup$


I need to find the value of



$$( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



I tried using the identity $a^{3} - b^{3}$ but couldn't reach very far and got stuck.






radicals







share|cite|improve this question











$endgroup$












  • $begingroup$
    What equation? What is there to solve?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:11










  • $begingroup$
    I mean to find its value.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:12










  • $begingroup$
    Edited the question. My bad.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:13










  • $begingroup$
    I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:14












  • $begingroup$
    Is there a specific form you are looking for in the final answer?
    $endgroup$
    – Aditya Dua
    Dec 12 '18 at 17:22














0












0








0


1



$begingroup$


I need to find the value of



$$( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



I tried using the identity $a^{3} - b^{3}$ but couldn't reach very far and got stuck.






radicals







share|cite|improve this question











$endgroup$




I need to find the value of



$$( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



I tried using the identity $a^{3} - b^{3}$ but couldn't reach very far and got stuck.






radicals




radicals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 17:13







navjotjsingh

















asked Dec 12 '18 at 17:11









navjotjsinghnavjotjsingh

1446




1446












  • $begingroup$
    What equation? What is there to solve?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:11










  • $begingroup$
    I mean to find its value.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:12










  • $begingroup$
    Edited the question. My bad.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:13










  • $begingroup$
    I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:14












  • $begingroup$
    Is there a specific form you are looking for in the final answer?
    $endgroup$
    – Aditya Dua
    Dec 12 '18 at 17:22


















  • $begingroup$
    What equation? What is there to solve?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:11










  • $begingroup$
    I mean to find its value.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:12










  • $begingroup$
    Edited the question. My bad.
    $endgroup$
    – navjotjsingh
    Dec 12 '18 at 17:13










  • $begingroup$
    I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
    $endgroup$
    – Servaes
    Dec 12 '18 at 17:14












  • $begingroup$
    Is there a specific form you are looking for in the final answer?
    $endgroup$
    – Aditya Dua
    Dec 12 '18 at 17:22
















$begingroup$
What equation? What is there to solve?
$endgroup$
– Servaes
Dec 12 '18 at 17:11




$begingroup$
What equation? What is there to solve?
$endgroup$
– Servaes
Dec 12 '18 at 17:11












$begingroup$
I mean to find its value.
$endgroup$
– navjotjsingh
Dec 12 '18 at 17:12




$begingroup$
I mean to find its value.
$endgroup$
– navjotjsingh
Dec 12 '18 at 17:12












$begingroup$
Edited the question. My bad.
$endgroup$
– navjotjsingh
Dec 12 '18 at 17:13




$begingroup$
Edited the question. My bad.
$endgroup$
– navjotjsingh
Dec 12 '18 at 17:13












$begingroup$
I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
$endgroup$
– Servaes
Dec 12 '18 at 17:14






$begingroup$
I'm guessing that $( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$ isn't the answer you're looking for? As that is the value. What kind of expression are you looking for?
$endgroup$
– Servaes
Dec 12 '18 at 17:14














$begingroup$
Is there a specific form you are looking for in the final answer?
$endgroup$
– Aditya Dua
Dec 12 '18 at 17:22




$begingroup$
Is there a specific form you are looking for in the final answer?
$endgroup$
– Aditya Dua
Dec 12 '18 at 17:22










2 Answers
2






active

oldest

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2












$begingroup$

Let
$$I=( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$
then
$$begin{aligned}
I^2 &=2sqrt{288}^3-2cdot 288sqrt{288-119}+6sqrt{288}cdot 119+2cdot 119sqrt{288-119} \
&= 2sqrt{288}^3-4394+714sqrt{288} \
&= 15480sqrt{2}-4394.
end{aligned}$$
Seeing as this can't be "simplified" further, I suspect the "neatest" form is then
$$I=sqrt{15480sqrt{2}-4394}.$$






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    Let $$x= (sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



    and $a= (sqrt{288} + sqrt{119})^{3/2}$ and $b= ( sqrt{288} - sqrt{119})^{3/2}$



    then $$a^2+b^2 = 2sqrt{288}^3 + 6sqrt{288}sqrt{119}^2 = 24sqrt{2}(288+3cdot 119)$$ $$ab = (288-119)^{3/2}= 13^3 =2197$$



    so $$x = sqrt{(a-b)^2} = sqrt{15480sqrt{2}-4394}$$






    share|cite|improve this answer









    $endgroup$













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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      Let
      $$I=( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$
      then
      $$begin{aligned}
      I^2 &=2sqrt{288}^3-2cdot 288sqrt{288-119}+6sqrt{288}cdot 119+2cdot 119sqrt{288-119} \
      &= 2sqrt{288}^3-4394+714sqrt{288} \
      &= 15480sqrt{2}-4394.
      end{aligned}$$
      Seeing as this can't be "simplified" further, I suspect the "neatest" form is then
      $$I=sqrt{15480sqrt{2}-4394}.$$






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        Let
        $$I=( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$
        then
        $$begin{aligned}
        I^2 &=2sqrt{288}^3-2cdot 288sqrt{288-119}+6sqrt{288}cdot 119+2cdot 119sqrt{288-119} \
        &= 2sqrt{288}^3-4394+714sqrt{288} \
        &= 15480sqrt{2}-4394.
        end{aligned}$$
        Seeing as this can't be "simplified" further, I suspect the "neatest" form is then
        $$I=sqrt{15480sqrt{2}-4394}.$$






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          Let
          $$I=( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$
          then
          $$begin{aligned}
          I^2 &=2sqrt{288}^3-2cdot 288sqrt{288-119}+6sqrt{288}cdot 119+2cdot 119sqrt{288-119} \
          &= 2sqrt{288}^3-4394+714sqrt{288} \
          &= 15480sqrt{2}-4394.
          end{aligned}$$
          Seeing as this can't be "simplified" further, I suspect the "neatest" form is then
          $$I=sqrt{15480sqrt{2}-4394}.$$






          share|cite|improve this answer









          $endgroup$



          Let
          $$I=( sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$
          then
          $$begin{aligned}
          I^2 &=2sqrt{288}^3-2cdot 288sqrt{288-119}+6sqrt{288}cdot 119+2cdot 119sqrt{288-119} \
          &= 2sqrt{288}^3-4394+714sqrt{288} \
          &= 15480sqrt{2}-4394.
          end{aligned}$$
          Seeing as this can't be "simplified" further, I suspect the "neatest" form is then
          $$I=sqrt{15480sqrt{2}-4394}.$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 12 '18 at 17:27









          Will FisherWill Fisher

          4,0381032




          4,0381032























              1












              $begingroup$

              Let $$x= (sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



              and $a= (sqrt{288} + sqrt{119})^{3/2}$ and $b= ( sqrt{288} - sqrt{119})^{3/2}$



              then $$a^2+b^2 = 2sqrt{288}^3 + 6sqrt{288}sqrt{119}^2 = 24sqrt{2}(288+3cdot 119)$$ $$ab = (288-119)^{3/2}= 13^3 =2197$$



              so $$x = sqrt{(a-b)^2} = sqrt{15480sqrt{2}-4394}$$






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                Let $$x= (sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



                and $a= (sqrt{288} + sqrt{119})^{3/2}$ and $b= ( sqrt{288} - sqrt{119})^{3/2}$



                then $$a^2+b^2 = 2sqrt{288}^3 + 6sqrt{288}sqrt{119}^2 = 24sqrt{2}(288+3cdot 119)$$ $$ab = (288-119)^{3/2}= 13^3 =2197$$



                so $$x = sqrt{(a-b)^2} = sqrt{15480sqrt{2}-4394}$$






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Let $$x= (sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



                  and $a= (sqrt{288} + sqrt{119})^{3/2}$ and $b= ( sqrt{288} - sqrt{119})^{3/2}$



                  then $$a^2+b^2 = 2sqrt{288}^3 + 6sqrt{288}sqrt{119}^2 = 24sqrt{2}(288+3cdot 119)$$ $$ab = (288-119)^{3/2}= 13^3 =2197$$



                  so $$x = sqrt{(a-b)^2} = sqrt{15480sqrt{2}-4394}$$






                  share|cite|improve this answer









                  $endgroup$



                  Let $$x= (sqrt{288} + sqrt{119})^{3/2} - ( sqrt{288} - sqrt{119})^{3/2}$$



                  and $a= (sqrt{288} + sqrt{119})^{3/2}$ and $b= ( sqrt{288} - sqrt{119})^{3/2}$



                  then $$a^2+b^2 = 2sqrt{288}^3 + 6sqrt{288}sqrt{119}^2 = 24sqrt{2}(288+3cdot 119)$$ $$ab = (288-119)^{3/2}= 13^3 =2197$$



                  so $$x = sqrt{(a-b)^2} = sqrt{15480sqrt{2}-4394}$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 12 '18 at 17:29









                  greedoidgreedoid

                  41.2k1150102




                  41.2k1150102






























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