Intuition problem.
I know 8^3 = 8*8*8
But, how would you write 8^1/3 in that format?
Intuition problem
Xavier Rogers
Daniel Sullivan
2*2*2
Benjamin Miller
How would you write 8^1/4 and 8^1/5?
Hudson Myers
2*2*2= 8
how would you format/write 8^1/3 so it equals with 2?
Bentley Myers
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
Henry Reed
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?
Christopher Kelly
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
Liam Gutierrez
ok, write it down
Bentley Moore
You don't. You add the following vectors (2,0,0),(0,2,0),(0,0,2) and calculate the longest possible diagonal.
Sebastian Morales
>You don't
NO WAY at all?
Owen Jones
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
Lucas Morgan
k^1/n * k^(n-1)/n = k
Christopher Murphy
>But, how would you write 8^1/3 in that format?
8 = (8^1/3)*(8^1/3)*(8^1/3)
Zachary Taylor
>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= ???
Ayden Hill
I'm not sure if I'm autistic enough to seriously reply to this thread.
Logan Evans
can you write in multiplication form why 8^1/3 is 2?
see:
Benjamin Sanchez
You could write 8/2/2=2.
But I see no reason for this exercise.
Owen Cruz
that's a division though, can you do it with multiplication? you know, because it's an exponent.
Nolan Cruz
8*(1/2)*(1/2)=2
Benjamin Hall
>8^1/3
>1/2
Something doesn't add up
Brody Walker
Are you autistic or something?
What are you even looking for, and why are you looking for it?
Evan Perry
>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"?
Gabriel Carter
>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.
Hudson Campbell
>you don't
Why the fuck not?
Is this somehow forbidden in math?
You should be able to, no?
Nathaniel Adams
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.
Eli Powell
Would you compare it to multiplying
Jeremiah Lee
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?
Logan Phillips
Yeah idk man like how is 2^2.5 what it is? I got nooo clue
Isaiah Diaz
Hmm
You mean that we cannot access "2" through 6/3 directly?
A long division solves that.
Xavier Perez
2^2.5= 2*2*0.5 = 2?
uhmm
Kevin Long
the short answer is, it's notation. 1/3 is the inverse of 3
want the long answer?
videos:
youtu.be
coursework, homework, etc:
wayback.archive-it.org
textbook:
Algebra, 2nd Edition, by Michael Artin
you seem autistic enough to get through it. godspeed user
Luis Murphy
>Intuition problem
You again.
You'll never develop any mathematical
intuition at this rate. It's developed by
long years of study, remember?
Brayden Walker
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.
Xavier Murphy
Fuck my latex up. What I wanted to write was:
[math](n^k)^\(\frac{1} {k} \) = n[/math]
Mason Watson
I give up.
[math](n^k)^\frac{1}{k} = n[/math]
Elijah Reed
This outlines a good way of thinking about it.