What are the possible values of a and b?

Question

kainui · Answer

$$\LARGE (x+1)+(x+a)^b = (x+1)(x+b)^a$$

gorv · Answer

x+1 will cancelled from both side

gorv · Answer

$$(x+a)^b=(x+b)^a$$

gorv · Answer

both will equal if and only if

gorv · Answer

1 their base are equal 
that is x+a=x+b 
2. their power are equal
that is a=b

kainui · Answer

How will they "cancel"?

gorv · Answer

we will divide bot side by x+1

kainui · Answer

Yeah, except it's added on the left side not multiplied, that won't work.

gorv · Answer

but you multiplied in your Q check that

kainui · Answer

Yeah I'm checking, you're wrong.

gorv · Answer

oohh sorry you said abt left side

kainui · Answer

Some things I'm noticing, if we do divide both sides by x+a we get $$\LARGE 1 + (x+a)^{b-1}=(x+b)^a$$

And there is this random example I found by trying stuff out. 

$$\LARGE (x+1)+(x+1)^2=(x+1)(x+2)$$

gorv · Answer

hmmm..try log

kainui · Answer

I made this problem up, so if someone wants to change something about it maybe we can figure out something interesting of something more general.

kainui · Answer

How about this instead, what numbers can we choose for a,b,c, and d: $$\LARGE (x+a)^b+(x+c)^d=(x+b)^a(x+d)^c$$

Obviously the example I chose we know a=a, b=1, c=2, d=1 is a working answer.

gorv · Answer

formula i have no idea ..but either go by random or solve