Intuition problem. I know 8^3 = 8*8*8 But, how would you write

2*2*2

>>9027171
How would you write 8^1/4 and 8^1/5?

>>9027171
2*2*2= 8
how would you format/write 8^1/3 so it equals with 2?

>>9027151
You wouldn't
Think of these properties as more fundamental of indices and then the repeated multiplication format makes sense, but the fractional indices also still make sense. They're not really how they're defined but it's better than to talk about repeated multiplication
x^n * x^m = x^(n + m)
(x^n)^m = x^(n * m)
x^0 = 1
x^1 = x

>>9027231
Wait, so you cannot write the 8^1/3 in a pure multiple form like:
2^3 = 2*2*2= 8

Sure, "indirectly" (root) it makes sense, but I try to conceptualize/visualize the fractional exponent in the traditional exponent multiplication format and I cant.

If you had to, how would you do it?

well, it is 8 multiplied with itself 1/3 times
now 1/3 is less than 1 so it must be less than 8
and since it's a third it makes sense that it would be a number which to the power of three makes eight

>>9027249
ok, write it down

>>9027151
You don't. You add the following vectors (2,0,0),(0,2,0),(0,0,2) and calculate the longest possible diagonal.

>>9027315
>You don't
NO WAY at all?

>>9027198
8^1/4*8^1/4*8^1/4*8^1/4 = 8 or
2^1/2*2^1/2*2^1/2*2^1/2 = 8
8^1/4 is an irrational number, so you might prefer just n*n*n*n = 8 or n^4 = 8

k^1/n * k^(n-1)/n = k

>>9027151
>But, how would you write 8^1/3 in that format?
8 = (8^1/3)*(8^1/3)*(8^1/3)

>>9028193
>8 = (8^1/3)*(8^1/3)*(8^1/3)
You solved it for 8 solve it for 2 now.

8^1/3=2
2= ???

I'm not sure if I'm autistic enough to seriously reply to this thread.

>>9029361
can you write in multiplication form why 8^1/3 is 2?

see:>>9029314

>>9029371
You could write 8/2/2=2.
But I see no reason for this exercise.

>>9029376
that's a division though, can you do it with multiplication? you know, because it's an exponent.

>>9029381
8*(1/2)*(1/2)=2

>>9029386
>8^1/3
>1/2
Something doesn't add up

>>9029418
Are you autistic or something?
What are you even looking for, and why are you looking for it?

>>9029424
>what are you looking for
Simple
2^3= 2*2*2 = 8
Exponentiation is about multiplying the number by itself.

8^1/3 = how do multiply 8 by itself 1/3 times so it gives off 2?
Your answer involved "1/2" and not "1/3".
How did you get the "1/2"?

>>9029431
>8^1/3 = how do multiply 8 by itself 1/3 times so it gives off 2?
You don't, you multiply 2 by itself 3 times to get 8. It's simply finding the reverse, like how dividing 8 by 2 is the same as multiplying it by 1/2.

>>9029439
>you don't
Why the fuck not?
Is this somehow forbidden in math?
You should be able to, no?

>>9029445
Because there's no way to write out what you're picturing, and also no reason to. It's not forbidden, it's just not how exponents actually work.

>>9029452
Would you compare it to multiplying <1 ?
E.g. 0.5*10 = 10/2 = 5

you can multiply 0.5 10 times but you cannot multiply 10 0.5 times, at least it wouldn't be """real"""" multiplication, it would be division essentially, even if you have the multiplication sign.

>>9027525
No, not really. The forward case of exponentiation, like 2^3 = 8 is known because its definition tells us 2*2*2 = 8. But that doesn't imply anything about the reverse case. It doesn't tell us how to arrive at 2 as the answer: we only know it's true because we study the forward case for so long.

You might not realize it but the same can be said of multiplication and division. 2*3 = 2+2+2 = 6, but 6/3 = 2 only because we know the forward case and we have a few ways of decomposing the number. 2*3 tells us exactly how we get to 6, but 6/3 doesn't necessarily tell us how to get back to 2. "Divide 6 into three equal groups" is the usual verbal description of simple whole division. Okay... how?

>>9027151
Yeah idk man like how is 2^2.5 what it is? I got nooo clue

>>9029537
Hmm
You mean that we cannot access "2" through 6/3 directly?
A long division solves that.

>>9029657
2^2.5= 2*2*0.5 = 2?
uhmm

>>9027151
the short answer is, it's notation. 1/3 is the inverse of 3

want the long answer?

videos:
https://youtu.be/VdLhQs_y_E8?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5

coursework, homework, etc:
http://wayback.archive-it.org/3671/20150528171650/https://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

textbook:
Algebra, 2nd Edition, by Michael Artin

you seem autistic enough to get through it. godspeed anon

>>9027151
>Intuition problem
You again.
You'll never develop any mathematical
intuition at this rate. It's developed by
long years of study, remember?

File: 1499773311353.png (58KB, 410x446px) Image search: [Google]

58KB, 410x446px

Serious answer - the "third root" is just an inverse operation of the cube.

[math](n^3)^1/3 = n[/math]

That's how it is defined. Basically what kind of root do you have to take to get back to the base number after exponentiation.

>>9030140
Fuck my latex up. What I wanted to write was:
[math](n^k)^\(\frac{1} {k} \) = n[/math]

>>9030148
I give up.

[math](n^k)^\frac{1}{k} = n[/math]

This outlines a good way of thinking about it.

https://www.youtube.com/watch?v=sULa9Lc4pck