Mixing
Is there supposed to be a difference between $x^{1/3}$ and $sqrt[3]{x}$ ? (Wolfram Alpha shows different...
$begingroup$
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
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
$endgroup$
add a comment |
$begingroup$
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
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
$endgroup$
1
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42
add a comment |
$begingroup$
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
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
$endgroup$
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
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
edited Jan 4 at 13:07
José Carlos Santos
155k22124227
155k22124227
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
asked Jan 4 at 12:37
RocketNutsRocketNuts
1254
1254
1
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42
add a comment |
1
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42
1
1
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
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votes
active
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votes
$begingroup$
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
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
$endgroup$
add a comment |
$begingroup$
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
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
$endgroup$
add a comment |
$begingroup$
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
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
$endgroup$
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
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
answered Jan 4 at 12:40
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
817110
817110
add a comment |
add a comment |
$begingroup$
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.
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
$endgroup$
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.
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
answered Jan 4 at 13:06
José Carlos SantosJosé Carlos Santos
155k22124227
155k22124227
add a comment |
add a comment |
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1
$begingroup$
I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics.
$endgroup$
– ajotatxe
Jan 4 at 12:42