This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it's impossible to define $\log(x)$ unambiguously in any reasonable way without further information. In the python module https://pypi.org/project/Octonion-Sedenion/, this problem is solved with some kind of short stack. This allows to define $\exp(x)$ and $\log(x)$ in such a way that always $\exp(\log(x)) = x$ and $\log(\exp(x)) = x$ both hold true for $x \in \mathbb{O}$ (octonions) and $x \in \mathbb{S}$ (sedenions). (In the same way it's possible to define these functions for complex $x \in \mathbb{C}$.)

With these definitions now it's possible to define the power function $x^y$ with the property $(x^y)^{1/y} = x$ at least for $x \in \mathbb{S}$ and $y \in \mathbb{O}$. (Since $\mathbb{S}$ is not alternative this no longer holds true for $y \in \mathbb{S}$.)

Question: Is there a reasonable interpretation of $x^y$ if $y \not\in \mathbb{R}_+$? $x^{0.5} = \sqrt{x}$ holds, further we have by specialisation f.i. $(x^{10})^{0.1} = x$. But what is the interpretation of $x^{3+4j}$ for octonions or sedenions $x$?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ unambiguously in any reasonable way without further information. In the python module https://pypi.org/project/Octonion-Sedenion/, this problem is solved with some kind of short stack. This allows to define $\exp(x)$ and $\log(x)$ in such a way that always $\exp(\log(x)) = x$ and $\log(\exp(x)) = x$ both hold true for $x \in \mathbb{O}$ (octonions) and $x \in \mathbb{S}$ (sedenions). (In the same way it is possible to define these functions for complex $x \in \mathbb{C}$.)

With these definitions now it is possible to define the power function $x^y$ with the property $(x^y)^{1/y} = x$ at least for $x \in \mathbb{S}$ and $y \in \mathbb{O}$. (Since $\mathbb{S}$ is not alternative this no longer holds true for $y \in \mathbb{S}$.)

Question: Is there a reasonable interpretation of $x^y$ if $y \not\in \mathbb{R}_+$? $x^{0.5} = \sqrt{x}$ holds, further we have by specialisation f.i. $(x^{10})^{0.1} = x$. But what is the interpretation of $x^{3+4j}$ for octonions or sedenions $x$?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$ its impossible to define $\log(x)$ unambiguously in any reasonable way without further information. In the python module https://pypi.org/project/Octonion-Sedenion/ this problem is solved with some kind of short stack. This allows to define $\exp(x)$ and $\log(x)$ in such a way that always $\exp(\log(x)) = x$ and $\log(\exp(x)) = x$ both hold true for $x \in \mathbb{O}$ (octonions) and $x \in \mathbb{S}$ (sedenions). (In the same way its possible to define these functions for complex $x \in \mathbb{C}$.

With these definitions now it's possible to define the power function $x^y$ with the property $(x^y)^{1/y} = x$ at least for $x \in \mathbb{S}$ and $y \in \mathbb{O}$. (Since $\mathbb{S}$ is not alternative this no longer holds true for $y \in \mathbb{S}$.)

Question: Is there a reasonable intepretation of $x^y$ if $y \not\in \mathbb{R}_+$? $x^{0.5} = \sqrt{x}$ holds, further we have by specialisation f.i. $(x^{10})^{0.1} = x$. But what is the interpretation of $x^{3+4j}$ for octonions or sedenions $x$?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it's impossible to define $\log(x)$ unambiguously in any reasonable way without further information. In the python module https://pypi.org/project/Octonion-Sedenion/, this problem is solved with some kind of short stack. This allows to define $\exp(x)$ and $\log(x)$ in such a way that always $\exp(\log(x)) = x$ and $\log(\exp(x)) = x$ both hold true for $x \in \mathbb{O}$ (octonions) and $x \in \mathbb{S}$ (sedenions). (In the same way it's possible to define these functions for complex $x \in \mathbb{C}$.)

With these definitions now it's possible to define the power function $x^y$ with the property $(x^y)^{1/y} = x$ at least for $x \in \mathbb{S}$ and $y \in \mathbb{O}$. (Since $\mathbb{S}$ is not alternative this no longer holds true for $y \in \mathbb{S}$.)

Question: Is there a reasonable interpretation of $x^y$ if $y \not\in \mathbb{R}_+$? $x^{0.5} = \sqrt{x}$ holds, further we have by specialisation f.i. $(x^{10})^{0.1} = x$. But what is the interpretation of $x^{3+4j}$ for octonions or sedenions $x$?