How can I transform exponential equation?

I try to transform following exponential equation. I would like to obtain an analytical equation for x:

>> syms P A B x y z n
>> eqn1= P== A*((x+y)^n)+B*(x+z)

eqn1 =

P == A*(x + y)^n + B*(x + z)

>> solve (eqn1,x)
Warning: Unable to find explicit solution. For options, see
help. 
> In solve (line 317) 

ans =

Empty sym: 0-by-1

Any ideas why MATLAB can not solve above code?

Answer 1

[John D'Errico]

Is there some reason why you think an analytical solution exists?

In fact, there is no analytical solution. Wanting magic to happen is not sufficient. How are your skills with a magic wand?

For example, suppose that n is greater than 4? There is a rather famous theorem showing that a polynomial equation of order greater than or equal to 5 has no analytical solution.

https://en.wikipedia.org/wiki/Abel–Ruffini_theorem

But in your case, n is completely unknown. n is a variable. So you have essentially a polynomial equation with non-constant coefficients, of completely unknown degree.

Only a crystal ball will suffice for you here.

[Walter Roberson]

Yup. Let A = 1, B = 2, n = 5, y = 0, z = 1, P = 0, so 0 = 1*(x + 0)^5 + 2*(x+1), so x^5 + 2*x + 2 = 0 . There are no closed form solutions to that.

[Konrad Dadej]

Thank you for answer - it is clear.

However, I wonder if the following assumptions would make it possible to solve? P>0 A>0 B>0 x>0 0<n<1 -1<y<1 -1<z<1 Is anybody can try to solve it?