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violinsane
Ciara
Artist
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.
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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.

Pf:

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   □
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:iconsculptor101:
sculptor101 Featured By Owner Feb 11, 2015  Professional Traditional Artist
Thanks so much for the watch!!
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:iconviolinsane:
violinsane Featured By Owner Feb 17, 2015
Np :) Your work is amazing!
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:iconsculptor101:
sculptor101 Featured By Owner Feb 18, 2015  Professional Traditional Artist
Huggle! Heart Heart Heart 
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:iconrevengerset:
revengerset Featured By Owner Sep 7, 2014  Hobbyist Photographer
Thanks for watching me :)
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:iconviolinsane:
violinsane Featured By Owner Sep 8, 2014
No prob; your work is really lovely :)
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:iconinspibrain101:
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...
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:iconviolinsane:
violinsane Featured By Owner Jul 1, 2014
:D
Ah...they are a well-kept secret ;)
: iconname : <--remove spaces
example :icondoctorwhoplz:
result: :icondoctorwhoplz: <-- oh hai!
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:iconnatpen:
natpen Featured By Owner May 25, 2014  Professional Digital Artist
thanks for the watch! :D
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:iconviolinsane:
violinsane Featured By Owner May 26, 2014
Absolutely! Your paintings are just beautiful; I can't wait to see more.
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:iconnatpen:
natpen Featured By Owner May 26, 2014  Professional Digital Artist
thanks! there will definitely more to come. :D 
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