Is there supposed to be a difference between $x^{1/3}$ and $sqrt[3]{x}$ ? (Wolfram Alpha shows different...












0














Compare these two functions:
plot $sqrt[3]{x}$
and
plot $x^{1/3}$



I understand how roots are ambiguous, and Wolfram Alpha apparently takes the principle root with the $x^{1/3}$ case and the real root with $sqrt[3]{x}$.



Is there any reason why the different approach? In the "input interpretation" it displays both as $sqrt[3]{x}$ and aren't they in fact supposed to mean the same? Isn't $sqrt[x]{y}$ defined as $y^{1/x}$ ?






complex-numbers roots graphing-functions radicals wolfram-alpha







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  • 1




    I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
    – ajotatxe
    Jan 4 at 12:42
















0














Compare these two functions:
plot $sqrt[3]{x}$
and
plot $x^{1/3}$



I understand how roots are ambiguous, and Wolfram Alpha apparently takes the principle root with the $x^{1/3}$ case and the real root with $sqrt[3]{x}$.



Is there any reason why the different approach? In the "input interpretation" it displays both as $sqrt[3]{x}$ and aren't they in fact supposed to mean the same? Isn't $sqrt[x]{y}$ defined as $y^{1/x}$ ?






complex-numbers roots graphing-functions radicals wolfram-alpha







share|cite|improve this question




















  • 1




    I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
    – ajotatxe
    Jan 4 at 12:42














0












0








0







Compare these two functions:
plot $sqrt[3]{x}$
and
plot $x^{1/3}$



I understand how roots are ambiguous, and Wolfram Alpha apparently takes the principle root with the $x^{1/3}$ case and the real root with $sqrt[3]{x}$.



Is there any reason why the different approach? In the "input interpretation" it displays both as $sqrt[3]{x}$ and aren't they in fact supposed to mean the same? Isn't $sqrt[x]{y}$ defined as $y^{1/x}$ ?






complex-numbers roots graphing-functions radicals wolfram-alpha







share|cite|improve this question















Compare these two functions:
plot $sqrt[3]{x}$
and
plot $x^{1/3}$



I understand how roots are ambiguous, and Wolfram Alpha apparently takes the principle root with the $x^{1/3}$ case and the real root with $sqrt[3]{x}$.



Is there any reason why the different approach? In the "input interpretation" it displays both as $sqrt[3]{x}$ and aren't they in fact supposed to mean the same? Isn't $sqrt[x]{y}$ defined as $y^{1/x}$ ?






complex-numbers roots graphing-functions radicals wolfram-alpha




complex-numbers roots graphing-functions radicals wolfram-alpha






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edited Jan 4 at 13:07









José Carlos Santos

152k22123226




152k22123226










asked Jan 4 at 12:37









RocketNutsRocketNuts

1254




1254








  • 1




    I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
    – ajotatxe
    Jan 4 at 12:42














  • 1




    I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
    – ajotatxe
    Jan 4 at 12:42








1




1




I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
– ajotatxe
Jan 4 at 12:42




I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
– ajotatxe
Jan 4 at 12:42










2 Answers
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It is taking $sqrt[3]x$ as the inverse of $x^3$, while $x^{frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph






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    0














    As far as Wolfram Alpha is concerned, $x^{frac13}$ is the principal cube root of $x$. Since that's not a real number when $x<0$, you can't see it in the graph.






    share|cite|improve this answer





















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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1














      It is taking $sqrt[3]x$ as the inverse of $x^3$, while $x^{frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph






      share|cite|improve this answer


























        1














        It is taking $sqrt[3]x$ as the inverse of $x^3$, while $x^{frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph






        share|cite|improve this answer
























          1












          1








          1






          It is taking $sqrt[3]x$ as the inverse of $x^3$, while $x^{frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph






          share|cite|improve this answer












          It is taking $sqrt[3]x$ as the inverse of $x^3$, while $x^{frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 4 at 12:40









          José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda

          807110




          807110























              0














              As far as Wolfram Alpha is concerned, $x^{frac13}$ is the principal cube root of $x$. Since that's not a real number when $x<0$, you can't see it in the graph.






              share|cite|improve this answer


























                0














                As far as Wolfram Alpha is concerned, $x^{frac13}$ is the principal cube root of $x$. Since that's not a real number when $x<0$, you can't see it in the graph.






                share|cite|improve this answer
























                  0












                  0








                  0






                  As far as Wolfram Alpha is concerned, $x^{frac13}$ is the principal cube root of $x$. Since that's not a real number when $x<0$, you can't see it in the graph.






                  share|cite|improve this answer












                  As far as Wolfram Alpha is concerned, $x^{frac13}$ is the principal cube root of $x$. Since that's not a real number when $x<0$, you can't see it in the graph.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 4 at 13:06









                  José Carlos SantosJosé Carlos Santos

                  152k22123226




                  152k22123226






























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