Mixing
Dividing power towers by exponents
Clash Royale CLAN TAG#URR8PPP
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
asked Jul 25 at 15:26
DonielF
409415
add a comment |Â
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
asked Jul 25 at 15:26
DonielF
409415
3
$e^4e = e^ee^ee^ee^e$.
– chepner
Jul 25 at 18:24
add a comment |Â
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
asked Jul 25 at 15:26
DonielF
409415
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
asked Jul 25 at 15:26
DonielF
409415
asked Jul 25 at 15:26
DonielF
409415
asked Jul 25 at 15:26
DonielF
409415
asked Jul 25 at 15:26
DonielF
409415
409415
3
$e^4e = e^ee^ee^ee^e$.
– chepner
Jul 25 at 18:24
add a comment |Â
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
add a comment |Â
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 $(a^b)^c=a^bc$.
edited Jul 26 at 0:45
Tanner Swett
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
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
add a comment |Â
up vote
6
down vote
Note that
$$e^e^e^e^eneq e^4e$$
as
$$3^3^3 = 3^27 neq 3^9$$
answered Jul 25 at 15:29
gimusi
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
add a comment |Â
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 $(a^b)^c=a^bc$.
edited Jul 26 at 0:45
Tanner Swett
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
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
add a comment |Â
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 $(a^b)^c=a^bc$.
edited Jul 26 at 0:45
Tanner Swett
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
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
add a comment |Â
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 $(a^b)^c=a^bc$.
edited Jul 26 at 0:45
Tanner Swett
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
33.5k136588
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 $(a^b)^c=a^bc$.
edited Jul 26 at 0:45
Tanner Swett
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
33.5k136588
edited Jul 26 at 0:45
Tanner Swett
3,6611536
edited Jul 26 at 0:45
Tanner Swett
3,6611536
edited Jul 26 at 0:45
Tanner Swett
3,6611536
3,6611536
answered Jul 25 at 15:28
Parcly Taxel
33.5k136588
answered Jul 25 at 15:28
Parcly Taxel
33.5k136588
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
add a comment |Â
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
add a comment |Â
up vote
6
down vote
Note that
$$e^e^e^e^eneq e^4e$$
as
$$3^3^3 = 3^27 neq 3^9$$
answered Jul 25 at 15:29
gimusi
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
add a comment |Â
up vote
6
down vote
Note that
$$e^e^e^e^eneq e^4e$$
as
$$3^3^3 = 3^27 neq 3^9$$
answered Jul 25 at 15:29
gimusi
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
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Note that
$$e^e^e^e^eneq e^4e$$
as
$$3^3^3 = 3^27 neq 3^9$$
answered Jul 25 at 15:29
gimusi
65k73583
Note that
$$e^e^e^e^eneq e^4e$$
as
$$3^3^3 = 3^27 neq 3^9$$
answered Jul 25 at 15:29
gimusi
65k73583
answered Jul 25 at 15:29
gimusi
65k73583
answered Jul 25 at 15:29
gimusi
65k73583
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
add a comment |Â
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
add a comment |Â
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3
$e^4e = e^ee^ee^ee^e$.
– chepner
Jul 25 at 18:24