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What is the reasoning behind this exponents question?
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What is $3^3^3?$
Plugging $3^3^3 $into the calculator gives 7625597484987.
I believe because this implies that
$3^3^3=3^27$, is this true?
And plugging $(3^3)^3$ gives 19683, because $
(3^3)^3=3^3times 3^3times 3^3=3^9=19683$
So which one is the correct answer, and why?
exponentiation
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
6
down vote
favorite
What is $3^3^3?$
Plugging $3^3^3 $into the calculator gives 7625597484987.
I believe because this implies that
$3^3^3=3^27$, is this true?
And plugging $(3^3)^3$ gives 19683, because $
(3^3)^3=3^3times 3^3times 3^3=3^9=19683$
So which one is the correct answer, and why?
exponentiation
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
1
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
What is $3^3^3?$
Plugging $3^3^3 $into the calculator gives 7625597484987.
I believe because this implies that
$3^3^3=3^27$, is this true?
And plugging $(3^3)^3$ gives 19683, because $
(3^3)^3=3^3times 3^3times 3^3=3^9=19683$
So which one is the correct answer, and why?
exponentiation
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is $3^3^3?$
Plugging $3^3^3 $into the calculator gives 7625597484987.
I believe because this implies that
$3^3^3=3^27$, is this true?
And plugging $(3^3)^3$ gives 19683, because $
(3^3)^3=3^3times 3^3times 3^3=3^9=19683$
So which one is the correct answer, and why?
exponentiation
exponentiation
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 48 mins ago
Meghan C
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 48 mins ago
Meghan C
383
asked 48 mins ago
Meghan C
383
383
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Meghan C is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
1
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago
add a comment |Â
2
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
1
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago
2
2
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
1
1
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
8
down vote
accepted
Unlike addition and multiplication, exponentiation is not associative:
- $(a+b)+c=a+(b+c)$
- $(atimes b)times c=atimes (btimes c)$
but
- ($a$^$b$)^$cne a!$^($b$^$c$), more commonly written as: $left(a^bright)^c ne a^left( b^c right)$
This means there's no risk in simply writing "$a+b+c$" or "$a times b times c$" since the order in which you perform the operations doesn't matter in both cases.
For exponentiation this is not the case and writing "$a!$^$b$^$c$" is ambiguous, but we do have:
$$colorblueleft(a^bright)^c = a^bc ne a^left( b^c right)$$
Because we have this property (in blue), it's common to interpret $a^b^c$ as $a^left( b^c right)$ but if you want to avoid confusion, you can always add the parentheses.
answered 36 mins ago
StackTD
20.6k1544
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
Unlike addition and multiplication, exponentiation is not associative:
- $(a+b)+c=a+(b+c)$
- $(atimes b)times c=atimes (btimes c)$
but
- ($a$^$b$)^$cne a!$^($b$^$c$), more commonly written as: $left(a^bright)^c ne a^left( b^c right)$
This means there's no risk in simply writing "$a+b+c$" or "$a times b times c$" since the order in which you perform the operations doesn't matter in both cases.
For exponentiation this is not the case and writing "$a!$^$b$^$c$" is ambiguous, but we do have:
$$colorblueleft(a^bright)^c = a^bc ne a^left( b^c right)$$
Because we have this property (in blue), it's common to interpret $a^b^c$ as $a^left( b^c right)$ but if you want to avoid confusion, you can always add the parentheses.
answered 36 mins ago
StackTD
20.6k1544
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
add a comment |Â
up vote
8
down vote
accepted
Unlike addition and multiplication, exponentiation is not associative:
- $(a+b)+c=a+(b+c)$
- $(atimes b)times c=atimes (btimes c)$
but
- ($a$^$b$)^$cne a!$^($b$^$c$), more commonly written as: $left(a^bright)^c ne a^left( b^c right)$
This means there's no risk in simply writing "$a+b+c$" or "$a times b times c$" since the order in which you perform the operations doesn't matter in both cases.
For exponentiation this is not the case and writing "$a!$^$b$^$c$" is ambiguous, but we do have:
$$colorblueleft(a^bright)^c = a^bc ne a^left( b^c right)$$
Because we have this property (in blue), it's common to interpret $a^b^c$ as $a^left( b^c right)$ but if you want to avoid confusion, you can always add the parentheses.
answered 36 mins ago
StackTD
20.6k1544
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
add a comment |Â
up vote
8
down vote
accepted
up vote
8
down vote
accepted
Unlike addition and multiplication, exponentiation is not associative:
- $(a+b)+c=a+(b+c)$
- $(atimes b)times c=atimes (btimes c)$
but
- ($a$^$b$)^$cne a!$^($b$^$c$), more commonly written as: $left(a^bright)^c ne a^left( b^c right)$
This means there's no risk in simply writing "$a+b+c$" or "$a times b times c$" since the order in which you perform the operations doesn't matter in both cases.
For exponentiation this is not the case and writing "$a!$^$b$^$c$" is ambiguous, but we do have:
$$colorblueleft(a^bright)^c = a^bc ne a^left( b^c right)$$
Because we have this property (in blue), it's common to interpret $a^b^c$ as $a^left( b^c right)$ but if you want to avoid confusion, you can always add the parentheses.
answered 36 mins ago
StackTD
20.6k1544
Unlike addition and multiplication, exponentiation is not associative:
- $(a+b)+c=a+(b+c)$
- $(atimes b)times c=atimes (btimes c)$
but
- ($a$^$b$)^$cne a!$^($b$^$c$), more commonly written as: $left(a^bright)^c ne a^left( b^c right)$
This means there's no risk in simply writing "$a+b+c$" or "$a times b times c$" since the order in which you perform the operations doesn't matter in both cases.
For exponentiation this is not the case and writing "$a!$^$b$^$c$" is ambiguous, but we do have:
$$colorblueleft(a^bright)^c = a^bc ne a^left( b^c right)$$
Because we have this property (in blue), it's common to interpret $a^b^c$ as $a^left( b^c right)$ but if you want to avoid confusion, you can always add the parentheses.
answered 36 mins ago
StackTD
20.6k1544
edited 5 mins ago
answered 36 mins ago
StackTD
20.6k1544
answered 36 mins ago
StackTD
20.6k1544
answered 36 mins ago
StackTD
20.6k1544
20.6k1544
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
add a comment |Â
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
What a thoughtful explanation. I really appreciate this!
– Meghan C
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
You're welcome, Meghan, and welcome to MSE.
– StackTD
12 mins ago
add a comment |Â
Meghan C is a new contributor. Be nice, and check out our Code of Conduct.
Meghan C is a new contributor. Be nice, and check out our Code of Conduct.
Meghan C is a new contributor. Be nice, and check out our Code of Conduct.
Meghan C is a new contributor. Be nice, and check out our Code of Conduct.
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2
People find the notation confusing, but $a^b^c$ means $a^(b^c)$ and not $(a^b)^c$, which of course is just $a^bc$. In practice, I think it's best to add the parentheses, as omitting them always leads to confusion.
– lulu
45 mins ago
1
You can interpret that there is no reason to read $3^3^3$ as $(3^3)^3$ because we can express it as $3^3cdot3$.
– Yves Daoust
29 mins ago
Thanks a lot. I really appreciate it.
– Meghan C
13 mins ago