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517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:27:13
To show G is cyclic
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:27:25
First try to construct the function $f$
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:27:28
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:27:39
I think it will be easier to map from <a> x <b> to G instead
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:28:01
Can we do that? Wouldn't it be assuming what we are proving?
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 19:28:05
no.
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:07
We construct f
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:28:08
Or to just get an idea?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:11
then prove it is an isomorphism
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:28:12
Oh
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:16
$$f : \langle a\rangle\times \langle b\rangle$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:28:19
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:32
$$f((x, y)) = ???$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 19:28:35
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:45
Then define where (x, y) must map to
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:52
for any (x, y) in <a> x <b>
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:28:59
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:29:12
A hint is --- you can write any generic (x, y) in a certain way
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 19:29:16
which makes defining f much easier.
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 19:29:20
Try it 🙂
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:34:45
I'm not sure what you meant by this
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 19:49:01
@Shuri2060 Shuri I'm still very confused
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:04:37
@Moon Child
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:04:43
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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ee650a1eb2184f5ca1221029aa080be4
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Shuri2060
02/05/2022 20:06:19
$=\{(e, e), (a, e), (a^2, e), \cdots, (a^m, e), \\ (e, b), (a, b), (a^2, b), \cdots, (a^m, b), \\ (e, b^2), (a, b^2), (a^2, b^2), \cdots, (a^m, b^2),\\ \vdots,\\ (e, b^n), (a, b^n), (a^2, b^n), \cdots, (a^m, b^n)\}$
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:06:35
ugly, one sec.
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ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 20:06:47
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:06:52
f((x,y)) = f(x^m)f(x^n) can we use order?
1
0
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:06:59
almost
0
1
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:07:07
Look how I have written <a> x <b>
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1
589,206,850,839,380,000
Moon Child
9119
Moon Child
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:07:11
it is almost like a matrix
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:07:23
All of the elements in there can be written in a specific form
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:08:42
You wrote it as (a,b)
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:09:11
everything in <a>
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:09:16
can be written a^k for some k
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:09:24
everything in <b> can be written b^j for some j
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:09:26
hint?
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:10:41
(a^k,b^j)
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:13:49
thats what i meant
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:14:28
Yeah, just clarifying that bit
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:15:13
I'm just not sure what you meant in that hint
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:50:23
?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:50:30
(a^k,b^j)
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:50:33
f((a^k,b^j))
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:50:40
You define where this maps to for every k and j
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:50:45
to define the function.
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:53:34
So in this case with the problem, can we say f((a^m,b^n))?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:53:51
m and n are constants.
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:53:54
a^m = e
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:53:59
b^n = e
0
1
589,206,850,839,380,000
Moon Child
9119
Moon Child
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:54:25
You need to specify where the function maps to for **every** k and j
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:54:26
f((a^k,b^j))
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:54:29
So we can't have that, which is why we are using x,y with k,j
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:54:36
not x, y
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:54:40
**a** and **b**
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:54:42
Well a,b
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:54:53
You should do the same as in the previous example
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:55:01
where should we send (a, e) to?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:55:07
f((a^1,b^0)) = ???
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:55:30
f((a^1,b^0)) = f((a, e)) = ???
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help-10
ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:56:06
(e,b)
1
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:56:16
no ???
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:56:20
let's get this clear
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:56:33
We need to create a bijection from **G** to <a> x <b>
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:56:45
The way I've written it, f maps starting from <a> <b>
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 20:57:00
Ohh sorry, I thought we were using the example since we stated it
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:57:19
$$f: \langle a\rangle\times\langle b\rangle\to G$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 20:57:22
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
517
help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:57:30
yh. I think it's probably easier to think the other way round
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:57:32
though for the general case
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:57:53
$$f((a^1,b^0)) = f((a, e)) = ???$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 20:57:57
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 20:58:05
So this must map to element in G
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 21:00:50
Which is a?
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 21:01:18
$$f((a^1,b^0)) = f((a, e)) = a = ae = a^1b^0$$
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help-10
ee650a1eb2184f5ca1221029aa080be4
510,789,298,321,096,700
TeXit
02/05/2022 21:01:38
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
0
1
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help-10
ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 21:01:45
I think you should be able to figure out what f should be
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ee650a1eb2184f5ca1221029aa080be4
321,445,119,171,756,000
Shuri2060
02/05/2022 21:01:49
Hint is there
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 21:03:09
f((a^k,b^j)) = a^kb^j
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ee650a1eb2184f5ca1221029aa080be4
589,206,850,839,380,000
Moon Child
02/05/2022 21:35:08
.close
1
0
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52447cfa779048a4a1e37f0ebef56e63
246,673,685,866,479,600
Illyria
02/06/2022 00:46:04
hello i don't seem to understand where my working went wrong
https://cdn.discordapp.c…0342/unknown.png
unknown.png
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
246,673,685,866,479,600
Illyria
02/06/2022 00:46:15
https://cdn.discordapp.c…0448/unknown.png
unknown.png
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
246,673,685,866,479,600
Illyria
02/06/2022 00:46:39
according to the markscheme the answer is supposed to be 1441π/20
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
246,673,685,866,479,600
Illyria
02/06/2022 00:47:08
by the way this is for question (a)
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
261,933,205,387,477,000
riemann
02/06/2022 02:01:15
@Ansh_ , @Random guy and I did almost an exact problem in #calculus
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1
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
261,933,205,387,477,000
riemann
02/06/2022 02:01:15
@Ansh_ , @Random guy and I did almost an exact problem in #calculus
0
1
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
261,933,205,387,477,000
riemann
02/06/2022 02:01:15
@Ansh_ , @Random guy and I did almost an exact problem in #calculus
0
1
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
673,062,964,059,701,200
Random guy
02/06/2022 02:02:48
yea thats similar
0
1
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518
help-10
52447cfa779048a4a1e37f0ebef56e63
246,673,685,866,479,600
Illyria
02/06/2022 02:03:55
Alright I will check
1
0
null
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:33:44
https://cdn.discordapp.c…0884/unknown.png
unknown.png
1
0
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:33:50
How do I express this surd with a rational denom
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519
help-10
317c81b6630f4cec952599379429da87
863,703,908,576,002,000
Solution
02/06/2022 03:34:50
Rationalize
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1
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:34:57
🧠
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:34:59
Great idea
1
0
706,934,233,012,371,600
Ansh_
3502
Arya
false
519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:35:04
Why haven't I thought of that
1
0
246,673,685,866,479,600
Illyria
8084
Illyria
false
519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:35:11
Please expand on your idea
1
0
673,062,964,059,701,200
Random guy
4029
Random guy
false
519
help-10
317c81b6630f4cec952599379429da87
863,703,908,576,002,000
Solution
02/06/2022 03:37:11
$\frac{1}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}$
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1
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519
help-10
317c81b6630f4cec952599379429da87
510,789,298,321,096,700
TeXit
02/06/2022 03:37:14
**BrownMunde**
https://cdn.discordapp.c…908576002061.png
863703908576002061.png
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1
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:37:21
Sounds rational
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519
help-10
317c81b6630f4cec952599379429da87
456,226,577,798,135,800
Deleted User
02/06/2022 03:38:09
Cheers for the help
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