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violinsane's Profile Picture
United States
I'm a Math-CS Major at university. When I'm not attempting to program something, I do freelance illustration and graphic design.
So, yesterday at work a student asked me why you can't distribute constants over parentheses when you have exponents (e.g. why doesn't a(x + y)2 = (ax + ay)2) and it got me thinking. Since a(x + y) = ax + ay, it's very tempting to just distribute blindly. So, what do you do if there are exponents involved? I couldn't find anything in the texts that we had there, so I came up with this; let me know what you think:

Let a, x, y ∈ ℝ, n ∈ ℕ. Then a(x + y)n = (a1/nx + a1/ny)n.


By the Binomial Theorem, (a1/nx + a1/ny)n = Σ nk=0 (nk) (a1/nx)n - k (a1/ny)k. Rewriting,
Σ nk=0 (nk) a1/n(n - k)xn - k a1/n(k)yk

= Σ nk=0 (nk) a(1 - k/n)xn - k ak/nyk

= Σ nk=0 (nk) a1 - k/n + k/nxn - kyk

= Σ nk=0 (nk) axn - kyk

= a Σ nk=0 (nk) xn - kyk

= a(x + y)n

So,  a(x + y)n = (a1/nx + a1/ny)n   □
  • Listening to: Tank!
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sculptor101 Featured By Owner Feb 11, 2015  Professional Traditional Artist
Thanks so much for the watch!!
violinsane Featured By Owner Feb 17, 2015
Np :) Your work is amazing!
sculptor101 Featured By Owner Feb 18, 2015  Professional Traditional Artist
Huggle! Heart Heart Heart 
revengerset Featured By Owner Sep 7, 2014  Hobbyist Photographer
Thanks for watching me :)
violinsane Featured By Owner Sep 8, 2014
No prob; your work is really lovely :)
inspibrain101 Featured By Owner Jun 24, 2014  Hobbyist General Artist
Awwww, thank you so much!
... I need to learn how to do the cool emoticons...
violinsane Featured By Owner Jul 1, 2014
Ah...they are a well-kept secret ;)
: iconname : <--remove spaces
example :icondoctorwhoplz:
result: :icondoctorwhoplz: <-- oh hai!
natpen Featured By Owner May 25, 2014  Professional Digital Artist
thanks for the watch! :D
violinsane Featured By Owner May 26, 2014
Absolutely! Your paintings are just beautiful; I can't wait to see more.
natpen Featured By Owner May 26, 2014  Professional Digital Artist
thanks! there will definitely more to come. :D 
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