What is $frac{1}{|{x}|}-frac{x^2}{|x|^3}$?











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What's the result of:
$$frac{1}{|{x}|}-frac{x^2}{|x|^3}$$
Is it
$$frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|x|}=0$$
or
$$frac{1}{|{x}|}-frac{x^2}{|x|^2x}=frac{1}{|{x}|}-frac{1}{x}frac{x^2}{|x|^2}=frac{1}{|{x}|}-frac{1}{x}=left{begin{matrix}frac{2}{x},x<0\ 0,x>0 end{matrix}right.$$










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  • 7




    First one is correct $|x|^2=x^2$
    – Yadati Kiran
    Nov 23 at 21:08






  • 4




    Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
    – celtschk
    Nov 23 at 22:17






  • 1




    Please also note that none of this would work with complex numbers ($xinmathbb C$).
    – Martin Rosenau
    Nov 24 at 13:38

















up vote
9
down vote

favorite
1












What's the result of:
$$frac{1}{|{x}|}-frac{x^2}{|x|^3}$$
Is it
$$frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|x|}=0$$
or
$$frac{1}{|{x}|}-frac{x^2}{|x|^2x}=frac{1}{|{x}|}-frac{1}{x}frac{x^2}{|x|^2}=frac{1}{|{x}|}-frac{1}{x}=left{begin{matrix}frac{2}{x},x<0\ 0,x>0 end{matrix}right.$$










share|cite|improve this question




















  • 7




    First one is correct $|x|^2=x^2$
    – Yadati Kiran
    Nov 23 at 21:08






  • 4




    Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
    – celtschk
    Nov 23 at 22:17






  • 1




    Please also note that none of this would work with complex numbers ($xinmathbb C$).
    – Martin Rosenau
    Nov 24 at 13:38















up vote
9
down vote

favorite
1









up vote
9
down vote

favorite
1






1





What's the result of:
$$frac{1}{|{x}|}-frac{x^2}{|x|^3}$$
Is it
$$frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|x|}=0$$
or
$$frac{1}{|{x}|}-frac{x^2}{|x|^2x}=frac{1}{|{x}|}-frac{1}{x}frac{x^2}{|x|^2}=frac{1}{|{x}|}-frac{1}{x}=left{begin{matrix}frac{2}{x},x<0\ 0,x>0 end{matrix}right.$$










share|cite|improve this question















What's the result of:
$$frac{1}{|{x}|}-frac{x^2}{|x|^3}$$
Is it
$$frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|x|}=0$$
or
$$frac{1}{|{x}|}-frac{x^2}{|x|^2x}=frac{1}{|{x}|}-frac{1}{x}frac{x^2}{|x|^2}=frac{1}{|{x}|}-frac{1}{x}=left{begin{matrix}frac{2}{x},x<0\ 0,x>0 end{matrix}right.$$







algebra-precalculus absolute-value






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edited Nov 24 at 13:36









user21820

38k541150




38k541150










asked Nov 23 at 21:07









BinaryBurst

356110




356110








  • 7




    First one is correct $|x|^2=x^2$
    – Yadati Kiran
    Nov 23 at 21:08






  • 4




    Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
    – celtschk
    Nov 23 at 22:17






  • 1




    Please also note that none of this would work with complex numbers ($xinmathbb C$).
    – Martin Rosenau
    Nov 24 at 13:38
















  • 7




    First one is correct $|x|^2=x^2$
    – Yadati Kiran
    Nov 23 at 21:08






  • 4




    Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
    – celtschk
    Nov 23 at 22:17






  • 1




    Please also note that none of this would work with complex numbers ($xinmathbb C$).
    – Martin Rosenau
    Nov 24 at 13:38










7




7




First one is correct $|x|^2=x^2$
– Yadati Kiran
Nov 23 at 21:08




First one is correct $|x|^2=x^2$
– Yadati Kiran
Nov 23 at 21:08




4




4




Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
– celtschk
Nov 23 at 22:17




Just insert $-1$. Which of the expressions are equal? (Note that this by itself is not a proof, as you have to rule out that both expressions are false, but it lets you quickly see that one of them simply cannot be right).
– celtschk
Nov 23 at 22:17




1




1




Please also note that none of this would work with complex numbers ($xinmathbb C$).
– Martin Rosenau
Nov 24 at 13:38






Please also note that none of this would work with complex numbers ($xinmathbb C$).
– Martin Rosenau
Nov 24 at 13:38












4 Answers
4






active

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up vote
17
down vote



accepted










As said in the comments, you have $|x|^2=x^2$ which is why the first proof is correct.



The error in the second proof is in the very beginning, namely $frac{1}{|{x}|}-frac{x^2}{|x|^2x}$ which is different than what you started with. It is wrong that $|x|^3=|x|^2x$. Take $x=-1$ for example.






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    up vote
    5
    down vote













    Your first conclusion is right since $$|x|^3=|x|^2cdot |x|=x^2cdot |x|$$and the second is wrong since for $x<0$ $$|x|^3=-x^3ne x^3= |x|^2cdot x$$






    share|cite|improve this answer




























      up vote
      2
      down vote













      Yes the first one is correct, indeed we have $|x|^3=|x||x^2|=|x|x^2$ then



      $$frac{1}{|{x}|}-frac{x^2}{|x|^3}=frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|{x}|}=0$$



      For the same reason the second one is wrong.






      share|cite|improve this answer




























        up vote
        1
        down vote













        As the other answers already told you, the first one is true.



        But maybe this way of thinking gives you an intuition: Both terms $frac{1}{|x|}$ and $frac{x²}{|x³|}$ are positive and obviously have the same absolute value, so since there is a minus in between, the result must be zero.






        share|cite|improve this answer





















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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          17
          down vote



          accepted










          As said in the comments, you have $|x|^2=x^2$ which is why the first proof is correct.



          The error in the second proof is in the very beginning, namely $frac{1}{|{x}|}-frac{x^2}{|x|^2x}$ which is different than what you started with. It is wrong that $|x|^3=|x|^2x$. Take $x=-1$ for example.






          share|cite|improve this answer

























            up vote
            17
            down vote



            accepted










            As said in the comments, you have $|x|^2=x^2$ which is why the first proof is correct.



            The error in the second proof is in the very beginning, namely $frac{1}{|{x}|}-frac{x^2}{|x|^2x}$ which is different than what you started with. It is wrong that $|x|^3=|x|^2x$. Take $x=-1$ for example.






            share|cite|improve this answer























              up vote
              17
              down vote



              accepted







              up vote
              17
              down vote



              accepted






              As said in the comments, you have $|x|^2=x^2$ which is why the first proof is correct.



              The error in the second proof is in the very beginning, namely $frac{1}{|{x}|}-frac{x^2}{|x|^2x}$ which is different than what you started with. It is wrong that $|x|^3=|x|^2x$. Take $x=-1$ for example.






              share|cite|improve this answer












              As said in the comments, you have $|x|^2=x^2$ which is why the first proof is correct.



              The error in the second proof is in the very beginning, namely $frac{1}{|{x}|}-frac{x^2}{|x|^2x}$ which is different than what you started with. It is wrong that $|x|^3=|x|^2x$. Take $x=-1$ for example.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Nov 23 at 21:14









              Scientifica

              6,06141332




              6,06141332






















                  up vote
                  5
                  down vote













                  Your first conclusion is right since $$|x|^3=|x|^2cdot |x|=x^2cdot |x|$$and the second is wrong since for $x<0$ $$|x|^3=-x^3ne x^3= |x|^2cdot x$$






                  share|cite|improve this answer

























                    up vote
                    5
                    down vote













                    Your first conclusion is right since $$|x|^3=|x|^2cdot |x|=x^2cdot |x|$$and the second is wrong since for $x<0$ $$|x|^3=-x^3ne x^3= |x|^2cdot x$$






                    share|cite|improve this answer























                      up vote
                      5
                      down vote










                      up vote
                      5
                      down vote









                      Your first conclusion is right since $$|x|^3=|x|^2cdot |x|=x^2cdot |x|$$and the second is wrong since for $x<0$ $$|x|^3=-x^3ne x^3= |x|^2cdot x$$






                      share|cite|improve this answer












                      Your first conclusion is right since $$|x|^3=|x|^2cdot |x|=x^2cdot |x|$$and the second is wrong since for $x<0$ $$|x|^3=-x^3ne x^3= |x|^2cdot x$$







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Nov 23 at 21:13









                      Mostafa Ayaz

                      12.4k3733




                      12.4k3733






















                          up vote
                          2
                          down vote













                          Yes the first one is correct, indeed we have $|x|^3=|x||x^2|=|x|x^2$ then



                          $$frac{1}{|{x}|}-frac{x^2}{|x|^3}=frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|{x}|}=0$$



                          For the same reason the second one is wrong.






                          share|cite|improve this answer

























                            up vote
                            2
                            down vote













                            Yes the first one is correct, indeed we have $|x|^3=|x||x^2|=|x|x^2$ then



                            $$frac{1}{|{x}|}-frac{x^2}{|x|^3}=frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|{x}|}=0$$



                            For the same reason the second one is wrong.






                            share|cite|improve this answer























                              up vote
                              2
                              down vote










                              up vote
                              2
                              down vote









                              Yes the first one is correct, indeed we have $|x|^3=|x||x^2|=|x|x^2$ then



                              $$frac{1}{|{x}|}-frac{x^2}{|x|^3}=frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|{x}|}=0$$



                              For the same reason the second one is wrong.






                              share|cite|improve this answer












                              Yes the first one is correct, indeed we have $|x|^3=|x||x^2|=|x|x^2$ then



                              $$frac{1}{|{x}|}-frac{x^2}{|x|^3}=frac{1}{|{x}|}-frac{x^2}{|x|x^2}=frac{1}{|{x}|}-frac{1}{|{x}|}=0$$



                              For the same reason the second one is wrong.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Nov 23 at 21:12









                              gimusi

                              87.7k74393




                              87.7k74393






















                                  up vote
                                  1
                                  down vote













                                  As the other answers already told you, the first one is true.



                                  But maybe this way of thinking gives you an intuition: Both terms $frac{1}{|x|}$ and $frac{x²}{|x³|}$ are positive and obviously have the same absolute value, so since there is a minus in between, the result must be zero.






                                  share|cite|improve this answer

























                                    up vote
                                    1
                                    down vote













                                    As the other answers already told you, the first one is true.



                                    But maybe this way of thinking gives you an intuition: Both terms $frac{1}{|x|}$ and $frac{x²}{|x³|}$ are positive and obviously have the same absolute value, so since there is a minus in between, the result must be zero.






                                    share|cite|improve this answer























                                      up vote
                                      1
                                      down vote










                                      up vote
                                      1
                                      down vote









                                      As the other answers already told you, the first one is true.



                                      But maybe this way of thinking gives you an intuition: Both terms $frac{1}{|x|}$ and $frac{x²}{|x³|}$ are positive and obviously have the same absolute value, so since there is a minus in between, the result must be zero.






                                      share|cite|improve this answer












                                      As the other answers already told you, the first one is true.



                                      But maybe this way of thinking gives you an intuition: Both terms $frac{1}{|x|}$ and $frac{x²}{|x³|}$ are positive and obviously have the same absolute value, so since there is a minus in between, the result must be zero.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Nov 24 at 10:18









                                      Fritz Hefter

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