1. **Stating the problem:** Solve the equation $$x^r y^r = x^r + y^r$$ for variables $x$, $y$, and $r$. 2. **Rewrite the equation:** Move all terms to one side: $$x^r y^r - x^r - y^r = 0$$ 3. **Factor the equation:** Notice that $x^r y^r = (xy)^r$, so rewrite as: $$ (xy)^r - x^r - y^r = 0 $$ 4. **Divide both sides by $x^r$ (assuming $x \neq 0$):** $$ \frac{(xy)^r}{x^r} - \frac{x^r}{x^r} - \frac{y^r}{x^r} = 0 \implies y^r - 1 - \left(\frac{y}{x}\right)^r = 0 $$ 5. **Rearrange:** $$ y^r - \left(\frac{y}{x}\right)^r = 1 $$ 6. **Rewrite the left side:** $$ y^r - y^r x^{-r} = y^r \left(1 - x^{-r}\right) = 1 $$ 7. **Express $y^r$:** $$ y^r = \frac{1}{1 - x^{-r}} = \frac{1}{1 - \frac{1}{x^r}} = \frac{x^r}{x^r - 1} $$ 8. **Take the $r$-th root to solve for $y$:** $$ y = \left(\frac{x^r}{x^r - 1}\right)^{\frac{1}{r}} $$ **Summary:** Given $x \neq 0$ and $x^r \neq 1$, the solution for $y$ in terms of $x$ and $r$ is $$ y = \left(\frac{x^r}{x^r - 1}\right)^{\frac{1}{r}} $$ This formula expresses $y$ explicitly, assuming the domain restrictions hold. **Note:** Special cases where $x^r = 1$ or $x=0$ need separate consideration.