How to make the nth root of a product act the same as simple multiplication in regard to parity?
I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors.
I've done some initial research, and reviews these Stack questions: Square roots — positive and negative and The Product Rule of Square Roots with Negative Numbers but I couldn't find the information I was seeking (or am not fully understanding the answers.)
Are the following expressions true? If not, how can I produce the those results?
$sqrt[2]{1*-1} = -1$
$sqrt[3]{1*1*-1} = -1$
$sqrt[3]{1*-1*-1} = 1$
[update] This is what the function does:
$sqrt[n]{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}} text{ }*text{ }
frac{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}}{Delta_1*Delta_2*...*Delta_n}$
such that if there are an odd number of negative factors, the product is negative, otherwise positive.
- Is there a more compact way to express this?
also, any tips on notation are appreciated.
notation roots radicals
add a comment |
I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors.
I've done some initial research, and reviews these Stack questions: Square roots — positive and negative and The Product Rule of Square Roots with Negative Numbers but I couldn't find the information I was seeking (or am not fully understanding the answers.)
Are the following expressions true? If not, how can I produce the those results?
$sqrt[2]{1*-1} = -1$
$sqrt[3]{1*1*-1} = -1$
$sqrt[3]{1*-1*-1} = 1$
[update] This is what the function does:
$sqrt[n]{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}} text{ }*text{ }
frac{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}}{Delta_1*Delta_2*...*Delta_n}$
such that if there are an odd number of negative factors, the product is negative, otherwise positive.
- Is there a more compact way to express this?
also, any tips on notation are appreciated.
notation roots radicals
$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04
add a comment |
I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors.
I've done some initial research, and reviews these Stack questions: Square roots — positive and negative and The Product Rule of Square Roots with Negative Numbers but I couldn't find the information I was seeking (or am not fully understanding the answers.)
Are the following expressions true? If not, how can I produce the those results?
$sqrt[2]{1*-1} = -1$
$sqrt[3]{1*1*-1} = -1$
$sqrt[3]{1*-1*-1} = 1$
[update] This is what the function does:
$sqrt[n]{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}} text{ }*text{ }
frac{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}}{Delta_1*Delta_2*...*Delta_n}$
such that if there are an odd number of negative factors, the product is negative, otherwise positive.
- Is there a more compact way to express this?
also, any tips on notation are appreciated.
notation roots radicals
I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors.
I've done some initial research, and reviews these Stack questions: Square roots — positive and negative and The Product Rule of Square Roots with Negative Numbers but I couldn't find the information I was seeking (or am not fully understanding the answers.)
Are the following expressions true? If not, how can I produce the those results?
$sqrt[2]{1*-1} = -1$
$sqrt[3]{1*1*-1} = -1$
$sqrt[3]{1*-1*-1} = 1$
[update] This is what the function does:
$sqrt[n]{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}} text{ }*text{ }
frac{overline{Delta_1}*overline{Delta_2} *...*overline{Delta_n}}{Delta_1*Delta_2*...*Delta_n}$
such that if there are an odd number of negative factors, the product is negative, otherwise positive.
- Is there a more compact way to express this?
also, any tips on notation are appreciated.
notation roots radicals
notation roots radicals
edited Nov 30 at 3:42
asked Nov 26 at 17:57
mnp
62
62
$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04
add a comment |
$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04
$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04
$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04
add a comment |
2 Answers
2
active
oldest
votes
We have that
$sqrt[2]{1times(-1)} = sqrt{-1} neq -1quad color{red}checkmark$ indeed $-1times -1=1$
$sqrt[3]{1times1times(-1)} = sqrt[3]{-1} = -1quad color{green}checkmark$ indeed $-1times -1times -1=-1$
$sqrt[3]{1times(-1)times(-1)} = sqrt[3]{1}=1quad color{green}checkmark$ indeed $1times 1times 1=1$
As a general rule
- for $nin mathbb{N}$ even and $age 0$ we have
$$sqrt[n] a=b iff bge 0 quad b^n=a$$
- for $nin mathbb{N}$ odd and we have
$$sqrt[n] a=b iff b^n=a$$
add a comment |
I am not sure if this is the method you want.
We have Euler's formula
$$e^{itheta} = isintheta+costheta$$
We can take $n^{th}$ root of both sides to obtain
$$sqrt[n]{e^{itheta}}=sqrt[n]{isintheta+costheta}tag{1}$$
It seems like you are only asking for the cases for $1$ and $-1$, so let's do the following.
It is clear that $sqrt[n]{1}$ is $1$ or $-1$ regardless of whether $n$ is odd or even. The problem is how to figure out $sqrt[n]{-1}$. For $n$ is odd, $sqrt[n]{-1}=-1$.
For $n$ is even, let's suppose $theta = pi$. Then, from $(1)$ we have
$$sqrt[n]{e^{ipi}}= sqrt[n]{-1}$$
$$e^{ifrac{pi}{n}} = sqrt[n]{-1}$$
$$sqrt[n]{-1}= isinfrac{pi}{n}+cosfrac{pi}{n}$$
I apologize if this is not what you are looking for.
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
We have that
$sqrt[2]{1times(-1)} = sqrt{-1} neq -1quad color{red}checkmark$ indeed $-1times -1=1$
$sqrt[3]{1times1times(-1)} = sqrt[3]{-1} = -1quad color{green}checkmark$ indeed $-1times -1times -1=-1$
$sqrt[3]{1times(-1)times(-1)} = sqrt[3]{1}=1quad color{green}checkmark$ indeed $1times 1times 1=1$
As a general rule
- for $nin mathbb{N}$ even and $age 0$ we have
$$sqrt[n] a=b iff bge 0 quad b^n=a$$
- for $nin mathbb{N}$ odd and we have
$$sqrt[n] a=b iff b^n=a$$
add a comment |
We have that
$sqrt[2]{1times(-1)} = sqrt{-1} neq -1quad color{red}checkmark$ indeed $-1times -1=1$
$sqrt[3]{1times1times(-1)} = sqrt[3]{-1} = -1quad color{green}checkmark$ indeed $-1times -1times -1=-1$
$sqrt[3]{1times(-1)times(-1)} = sqrt[3]{1}=1quad color{green}checkmark$ indeed $1times 1times 1=1$
As a general rule
- for $nin mathbb{N}$ even and $age 0$ we have
$$sqrt[n] a=b iff bge 0 quad b^n=a$$
- for $nin mathbb{N}$ odd and we have
$$sqrt[n] a=b iff b^n=a$$
add a comment |
We have that
$sqrt[2]{1times(-1)} = sqrt{-1} neq -1quad color{red}checkmark$ indeed $-1times -1=1$
$sqrt[3]{1times1times(-1)} = sqrt[3]{-1} = -1quad color{green}checkmark$ indeed $-1times -1times -1=-1$
$sqrt[3]{1times(-1)times(-1)} = sqrt[3]{1}=1quad color{green}checkmark$ indeed $1times 1times 1=1$
As a general rule
- for $nin mathbb{N}$ even and $age 0$ we have
$$sqrt[n] a=b iff bge 0 quad b^n=a$$
- for $nin mathbb{N}$ odd and we have
$$sqrt[n] a=b iff b^n=a$$
We have that
$sqrt[2]{1times(-1)} = sqrt{-1} neq -1quad color{red}checkmark$ indeed $-1times -1=1$
$sqrt[3]{1times1times(-1)} = sqrt[3]{-1} = -1quad color{green}checkmark$ indeed $-1times -1times -1=-1$
$sqrt[3]{1times(-1)times(-1)} = sqrt[3]{1}=1quad color{green}checkmark$ indeed $1times 1times 1=1$
As a general rule
- for $nin mathbb{N}$ even and $age 0$ we have
$$sqrt[n] a=b iff bge 0 quad b^n=a$$
- for $nin mathbb{N}$ odd and we have
$$sqrt[n] a=b iff b^n=a$$
edited Nov 26 at 18:10
answered Nov 26 at 18:04
gimusi
1
1
add a comment |
add a comment |
I am not sure if this is the method you want.
We have Euler's formula
$$e^{itheta} = isintheta+costheta$$
We can take $n^{th}$ root of both sides to obtain
$$sqrt[n]{e^{itheta}}=sqrt[n]{isintheta+costheta}tag{1}$$
It seems like you are only asking for the cases for $1$ and $-1$, so let's do the following.
It is clear that $sqrt[n]{1}$ is $1$ or $-1$ regardless of whether $n$ is odd or even. The problem is how to figure out $sqrt[n]{-1}$. For $n$ is odd, $sqrt[n]{-1}=-1$.
For $n$ is even, let's suppose $theta = pi$. Then, from $(1)$ we have
$$sqrt[n]{e^{ipi}}= sqrt[n]{-1}$$
$$e^{ifrac{pi}{n}} = sqrt[n]{-1}$$
$$sqrt[n]{-1}= isinfrac{pi}{n}+cosfrac{pi}{n}$$
I apologize if this is not what you are looking for.
add a comment |
I am not sure if this is the method you want.
We have Euler's formula
$$e^{itheta} = isintheta+costheta$$
We can take $n^{th}$ root of both sides to obtain
$$sqrt[n]{e^{itheta}}=sqrt[n]{isintheta+costheta}tag{1}$$
It seems like you are only asking for the cases for $1$ and $-1$, so let's do the following.
It is clear that $sqrt[n]{1}$ is $1$ or $-1$ regardless of whether $n$ is odd or even. The problem is how to figure out $sqrt[n]{-1}$. For $n$ is odd, $sqrt[n]{-1}=-1$.
For $n$ is even, let's suppose $theta = pi$. Then, from $(1)$ we have
$$sqrt[n]{e^{ipi}}= sqrt[n]{-1}$$
$$e^{ifrac{pi}{n}} = sqrt[n]{-1}$$
$$sqrt[n]{-1}= isinfrac{pi}{n}+cosfrac{pi}{n}$$
I apologize if this is not what you are looking for.
add a comment |
I am not sure if this is the method you want.
We have Euler's formula
$$e^{itheta} = isintheta+costheta$$
We can take $n^{th}$ root of both sides to obtain
$$sqrt[n]{e^{itheta}}=sqrt[n]{isintheta+costheta}tag{1}$$
It seems like you are only asking for the cases for $1$ and $-1$, so let's do the following.
It is clear that $sqrt[n]{1}$ is $1$ or $-1$ regardless of whether $n$ is odd or even. The problem is how to figure out $sqrt[n]{-1}$. For $n$ is odd, $sqrt[n]{-1}=-1$.
For $n$ is even, let's suppose $theta = pi$. Then, from $(1)$ we have
$$sqrt[n]{e^{ipi}}= sqrt[n]{-1}$$
$$e^{ifrac{pi}{n}} = sqrt[n]{-1}$$
$$sqrt[n]{-1}= isinfrac{pi}{n}+cosfrac{pi}{n}$$
I apologize if this is not what you are looking for.
I am not sure if this is the method you want.
We have Euler's formula
$$e^{itheta} = isintheta+costheta$$
We can take $n^{th}$ root of both sides to obtain
$$sqrt[n]{e^{itheta}}=sqrt[n]{isintheta+costheta}tag{1}$$
It seems like you are only asking for the cases for $1$ and $-1$, so let's do the following.
It is clear that $sqrt[n]{1}$ is $1$ or $-1$ regardless of whether $n$ is odd or even. The problem is how to figure out $sqrt[n]{-1}$. For $n$ is odd, $sqrt[n]{-1}=-1$.
For $n$ is even, let's suppose $theta = pi$. Then, from $(1)$ we have
$$sqrt[n]{e^{ipi}}= sqrt[n]{-1}$$
$$e^{ifrac{pi}{n}} = sqrt[n]{-1}$$
$$sqrt[n]{-1}= isinfrac{pi}{n}+cosfrac{pi}{n}$$
I apologize if this is not what you are looking for.
answered Nov 26 at 18:43
Larry
1,6082722
1,6082722
add a comment |
add a comment |
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$sqrt{-1} =i$, so the first statement is wrong.
– Larry
Nov 26 at 18:04