How to re-write one fraction as two others. I have the two following fractions. <mstyle displa

Micah Haynes

Micah Haynes

Answered question

2022-05-16

How to re-write one fraction as two others.
I have the two following fractions.
A B x α + 1
and
C D x α + β
The form i want
E F x α + β + 1
I was thinking to do partial fractions or potentially doing a Laplace or Fourier transform, then rearranging and transforming back. How could one possibly get the two fractions into one fraction of this form.\ A,B,C,D,E and F can be number or possible a factorial if we go down the laplace transform route, as can α,β

Answer & Explanation

taweirrvb

taweirrvb

Beginner2022-05-17Added 21 answers

I'm assuming, as you seem to have implied, that A , B , C , D , E , F are arbitrary, and, in particular, that E and F can be defined as some combination of A , B , C , D
A B x α + 1 = A B x α + 1 x β x β = A x β B x α + 1 x β = A x β B x α + β + 1
And, similarly,
C D x α + β = C x D x α + β + 1
So you can have
A x β B x α + β + 1 C x D x α + β + 1 x β 1 = A x β B x α + β + 1 C x β D x α + β + 1 = A D B C B D x α + β + 1
and thus, we can let E = A D B C and F = B D to get
E F x α + β + 1 = ( A B x α + 1 ) ( C D x α + β ) x β 1
Edit:
This also shows that you cannot get appropriate values of E,F from a linear combination. Suppose, for the sake of contradiction, that there are constants M,N such that
M ( A B x α + 1 ) + N ( C D x α + β ) = E 1 F 1 x α + β + 1
for some E 1 , F 1 Then we have
E / E 1 F / F 1 ( M ( A B x α + 1 ) + N ( C D x α + β ) ) = E / E 1 F / F 1 E 1 F 1 x α + β + 1 = E F x α + β + 1
For simplicity, let ϵ = E / E 1 F / F 1 . Then,
ϵ M ( A B x α + 1 ) + ϵ N ( C D x α + β ) = ( A B x α + 1 ) ( C D x α + β ) x β 1 ϵ M ( A B x α + 1 ) = ( A B x α + 1 ) ( C D x α + β ) ( x β 1 + ϵ N ) ϵ M = 1 ( B C A D x β 1 ) ( x β 1 + ϵ N ) ϵ M = 1 ϵ B C N A D x β 1 M = 1 ϵ B C N A D x β 1
And this contradicts the assumption that M is constant.
Spencer Lutz

Spencer Lutz

Beginner2022-05-18Added 6 answers

Assuming that I properly understood (which is not sure, I must confess) : let
P = A B x α + 1
Q = C D x α + β
So
P 1 α + 1 = ( A B ) 1 α + 1 1 x
P 1 α + 1 Q = ( A B ) 1 α + 1 ( C D ) 1 x α + β + 1 = E F x α + β + 1
So
E F = ( A B ) 1 α + 1 ( C D )
You can choose any value you want for E or F and compute the other one from the above equation.

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