SOLUTION: a/b=c/d=e/f, a^3+c^3+e^3/b^3+d^3+f^3=?

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Question 1086326: a/b=c/d=e/f, a^3+c^3+e^3/b^3+d^3+f^3=?
Found 2 solutions by AnlytcPhil, ikleyn:
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
a%2Fb=c%2Fd=e%2Ff let each of those equal to k. Then

a=bk, c=dk, e=fk

%28%28bk%29%5E3%2B%28dk%29%5E3%2B%28fk%29%5E3%29%2F%28b%5E3%2Bd%5E3%2Bf%5E3%29

%28b%5E3k%5E3%2Bd%5E3k%5E3%2Bf%5E3k%5E3%29%2F%28b%5E3%2Bd%5E3%2Bf%5E3%29

%28k%5E3%28b%5E3%2Bd%5E3%2Bf%5E3%29%29%2F%28b%5E3%2Bd%5E3%2Bf%5E3%29

%28k%5E3%28cross%28b%5E3%2Bd%5E3%2Bf%5E3%29%29%29%2F%28cross%28b%5E3%2Bd%5E3%2Bf%5E3%29%29

k%5E3

So 

%28a%5E3%2Bc%5E3%2Be%5E3%29%2F%28b%5E3%2Bd%5E3%2Bf%5E3%29%22%22=%22%22k%5E3%22%22=%22%22a%5E3%2Fb%5E3%22%22=%22%22c%5E3%2Fd%5E3%22%22=%22%22e%5E3%2Ff%5E3

Edwin

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

From Wikipedia article https://en.wikipedia.org/wiki/Lemma_(mathematics) 
    In mathematics, a  lemma (a "helping theorem") is a proved proposition which is used as a stepping stone 
    to a larger result rather than as a statement of interest by itself. 


Lemma 1.  

    If a%2Fb = c%2Fd then %28a+%2B+c%29%2F%28b%2Bd%29 = a%2Fb = c%2Fd.


        Proof. Let a%2Fb = c%2Fd = k. Then

               a = b*k,  c = d*k  and  %28a+%2B+c%29%2F%28b%2Bd%29 = %28bk+%2B+dk%29%2F%28b+%2B+d%29 = %28%28b%2Bd%29%2Ak%29%2F%28b%2Bd%29 = k = a%2Fb = c%2Fd.

        The lemma is proved.



Lemma 2. 

    If a%2Fb = c%2Fd = e%2Ff  then  a%5E3%2Fb%5E3 = c%5E3%2Fd%5E3 = e%5E3%2Fv%5E3.


       The proof is obvious.



Major statement.  %28a%5E3%2Bc%5E3%2Be%5E3%29%2F%28b%5E3%2Bd%5E3%2Bf%5E3%29 = %28a%2Fb%29%5E3.


    Proof. It follows IMMEDIATELY Lemma 1  and  Lemma 2.