Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

Given,

$p=3^w,q=3^x,r=3^y,s=3^z$ and $p^2+q^3+r^5=s^7$

So, $3^{2w}+3^{3x}+3^{5y}=3^{7z}$

Let, $a=2w,b=3x,c=5y,d=7z$

So $3^a+3^b+3^c=3^d$

Without loss of generality, we assume $a \leq b \leq c $

Then $3^a(1+3^{b-a}+3^{c-a})=3^d$

Now right hand side is a power of $3$.

So no other prime than $3$ divides left hand side

So $a=b=c$

So $3 \times 3^a=3^d$

$3^{a+1}=3^d$

So $a+1=d$

So $2w=3x=5y=7z-1$

To find the minimum value of $w+x+y+z$,

We have to find the minimum number $n$ such n is divisible by $2,3,5$ and $n+1$ is divisible by $7$

We find such $n$ is $90$.

So $w=45,x=30,y=18,z=13$

So, minimum $w+x+y+z=45+30+18+13=106$