Dividing power towers by exponents

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up vote
2
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Say we have $e^e^e^e^e$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^4e$:



$$e^e^e^e^e=e^4e$$



Factoring out an $e^e$:



$$e^e^e^e=e^4$$



The left side now collapses to $e^3e$, leading to the equality $3e=4$.



Clearly my mistake was in the part where I divided both sides by $e^e$, but why am I not able to do this?




algebra-precalculus power-towers




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  • 3




    $e^4e = e^ee^ee^ee^e$.
    – chepner
    Jul 25 at 18:24















up vote
2
down vote

favorite












Say we have $e^e^e^e^e$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^4e$:



$$e^e^e^e^e=e^4e$$



Factoring out an $e^e$:



$$e^e^e^e=e^4$$



The left side now collapses to $e^3e$, leading to the equality $3e=4$.



Clearly my mistake was in the part where I divided both sides by $e^e$, but why am I not able to do this?




algebra-precalculus power-towers




share|cite|improve this question















  • 3




    $e^4e = e^ee^ee^ee^e$.
    – chepner
    Jul 25 at 18:24













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Say we have $e^e^e^e^e$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^4e$:



$$e^e^e^e^e=e^4e$$



Factoring out an $e^e$:



$$e^e^e^e=e^4$$



The left side now collapses to $e^3e$, leading to the equality $3e=4$.



Clearly my mistake was in the part where I divided both sides by $e^e$, but why am I not able to do this?




algebra-precalculus power-towers




share|cite|improve this question











Say we have $e^e^e^e^e$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^4e$:



$$e^e^e^e^e=e^4e$$



Factoring out an $e^e$:



$$e^e^e^e=e^4$$



The left side now collapses to $e^3e$, leading to the equality $3e=4$.



Clearly my mistake was in the part where I divided both sides by $e^e$, but why am I not able to do this?





algebra-precalculus power-towers





share|cite|improve this question










share|cite|improve this question




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asked Jul 25 at 15:26









DonielF

409415




409415







  • 3




    $e^4e = e^ee^ee^ee^e$.
    – chepner
    Jul 25 at 18:24













  • 3




    $e^4e = e^ee^ee^ee^e$.
    – chepner
    Jul 25 at 18:24








3




3




$e^4e = e^ee^ee^ee^e$.
– chepner
Jul 25 at 18:24





$e^4e = e^ee^ee^ee^e$.
– chepner
Jul 25 at 18:24











2 Answers
2






active

oldest

votes

















up vote
13
down vote



accepted










It's because $e^e^e^e^e$ isn't $e^4e$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^e^4$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^bc$.






share|cite|improve this answer



















  • 5




    $(((e^e)^e)^e)^e$ also isn't $e^4e$
    – Omnomnomnom
    Jul 25 at 15:29










  • Also note the similarity in the MathJax: e^e^e^e^e
    – Simply Beautiful Art
    44 mins ago


















up vote
6
down vote













Note that



$$e^e^e^e^eneq e^4e$$



as



$$3^3^3 = 3^27 neq 3^9$$






share|cite|improve this answer

















  • 1




    "Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
    – Jeppe Stig Nielsen
    Jul 25 at 20:23










  • @JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
    – gimusi
    Jul 25 at 20:35










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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
13
down vote



accepted










It's because $e^e^e^e^e$ isn't $e^4e$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^e^4$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^bc$.






share|cite|improve this answer



















  • 5




    $(((e^e)^e)^e)^e$ also isn't $e^4e$
    – Omnomnomnom
    Jul 25 at 15:29










  • Also note the similarity in the MathJax: e^e^e^e^e
    – Simply Beautiful Art
    44 mins ago















up vote
13
down vote



accepted










It's because $e^e^e^e^e$ isn't $e^4e$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^e^4$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^bc$.






share|cite|improve this answer



















  • 5




    $(((e^e)^e)^e)^e$ also isn't $e^4e$
    – Omnomnomnom
    Jul 25 at 15:29










  • Also note the similarity in the MathJax: e^e^e^e^e
    – Simply Beautiful Art
    44 mins ago













up vote
13
down vote



accepted







up vote
13
down vote



accepted






It's because $e^e^e^e^e$ isn't $e^4e$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^e^4$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^bc$.






share|cite|improve this answer















It's because $e^e^e^e^e$ isn't $e^4e$ in the first place (and neither is $((((e^e)^e)^e)^e)$, which is $e^e^4$). The left expression is a power tower, where exponentiation happens right-to-left, not left-to-right as required for the power multiplication rule $(a^b)^c=a^bc$.







share|cite|improve this answer















share|cite|improve this answer



share|cite|improve this answer








edited Jul 26 at 0:45









Tanner Swett

3,6611536




3,6611536











answered Jul 25 at 15:28









Parcly Taxel

33.5k136588




33.5k136588







  • 5




    $(((e^e)^e)^e)^e$ also isn't $e^4e$
    – Omnomnomnom
    Jul 25 at 15:29










  • Also note the similarity in the MathJax: e^e^e^e^e
    – Simply Beautiful Art
    44 mins ago













  • 5




    $(((e^e)^e)^e)^e$ also isn't $e^4e$
    – Omnomnomnom
    Jul 25 at 15:29










  • Also note the similarity in the MathJax: e^e^e^e^e
    – Simply Beautiful Art
    44 mins ago








5




5




$(((e^e)^e)^e)^e$ also isn't $e^4e$
– Omnomnomnom
Jul 25 at 15:29




$(((e^e)^e)^e)^e$ also isn't $e^4e$
– Omnomnomnom
Jul 25 at 15:29












Also note the similarity in the MathJax: e^e^e^e^e
– Simply Beautiful Art
44 mins ago





Also note the similarity in the MathJax: e^e^e^e^e
– Simply Beautiful Art
44 mins ago











up vote
6
down vote













Note that



$$e^e^e^e^eneq e^4e$$



as



$$3^3^3 = 3^27 neq 3^9$$






share|cite|improve this answer

















  • 1




    "Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
    – Jeppe Stig Nielsen
    Jul 25 at 20:23










  • @JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
    – gimusi
    Jul 25 at 20:35














up vote
6
down vote













Note that



$$e^e^e^e^eneq e^4e$$



as



$$3^3^3 = 3^27 neq 3^9$$






share|cite|improve this answer

















  • 1




    "Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
    – Jeppe Stig Nielsen
    Jul 25 at 20:23










  • @JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
    – gimusi
    Jul 25 at 20:35












up vote
6
down vote










up vote
6
down vote









Note that



$$e^e^e^e^eneq e^4e$$



as



$$3^3^3 = 3^27 neq 3^9$$






share|cite|improve this answer













Note that



$$e^e^e^e^eneq e^4e$$



as



$$3^3^3 = 3^27 neq 3^9$$







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 25 at 15:29









gimusi

65k73583




65k73583







  • 1




    "Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
    – Jeppe Stig Nielsen
    Jul 25 at 20:23










  • @JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
    – gimusi
    Jul 25 at 20:35












  • 1




    "Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
    – Jeppe Stig Nielsen
    Jul 25 at 20:23










  • @JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
    – gimusi
    Jul 25 at 20:35







1




1




"Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
– Jeppe Stig Nielsen
Jul 25 at 20:23




"Note that $$2^4 ne 4^2$$ as $$9^3 = (3^2)^3 = 3^6 ne 3^9$$ right?" Sorry, just teasing; I know how your answer is to be understood. You are saying that $x^x^x^x^x = x^4x$ is not a valid identity for all $x$ (because there is no power rule of the type the asker seems to use), so it would be strange if it was valid for $x=e$. However, isolated solutions like $x=1.69447ldots$ exist.
– Jeppe Stig Nielsen
Jul 25 at 20:23












@JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
– gimusi
Jul 25 at 20:35




@JeppeStigNielsen We have that $2^4=(2^2)^2=4^2=16$ and yes we are claiming that $x^x^x^x^x = x^4x$ is not true in general and in particular also for $x=e$.
– gimusi
Jul 25 at 20:35












 

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