If $x^a + x^b = x^c + x^d$ how do $a ,b , c , d$ relationship are?

Question

I used to solved these equation style and it's accidentally found an answer from matching $a, b, c,$ and $d$ relationship

when $x^a + x^b = x^c + x^d $ (I assume that $ab = cd$)

and found that's was wrong

therefore what's the true relationship between $a, b, c,$ and $d$ ?

For instance: $$3^{(3x^2 +8)} + 3^{(4x +2)} = 3^0 + 3^{(5x^2 +7)}$$ and $$(3x^2 +8)(4x +2) \not = 0 $$

Answer 1

If I understand you than your example is wrong . For integer values of a,b,c,d,x(x>1) we can say that $min(a,b)=min(c,d),max(a,b)=max(c,d) $.
Prove : Let $a<=b$ and $c<=d$ ; then we write $$x^b(x^{a-b}-1)=x^d(x^{c-d}-1)$$Also $(x,x^{a-b}-1)=1$ and $(x,x^{c-d}-1)=1$. $x^b|x^b(x^{a-b}-1)$ than we can say that $x^b|x^d(x^{c-d}-1)$ and $x^b|x^d$ .The same way we can pwrove that $x^d|x^b$ . Than we get that $b=d$ . So we get $$x^b(x^{a-b}-1)=x^b(x^{c-d}-1)$$ $$x^{a-b}-1=x^{c-d}-1$$From this equation we can write $a-b=c-d$ .Using that $b=d$ we have that $a=c$.