1. **Stating the problem:** Solve the equation $$2x^r - ax^r - v = 0$$ for $x^r$ and find $x$. 2. **Rewrite the equation:** Let $$t = x^r$$, then the equation becomes $$2t - a t - v = 0$$ or equivalently $$ (2 - a) t = v$$. 3. **Solve for $t$:** $$t = \frac{v}{2 - a}$$ 4. **Find $x$ from $t$:** Since $$t = x^r$$, we have $$x = \pm \sqrt[r]{t} = \pm \sqrt[r]{\frac{v}{2 - a}}$$ 5. **Summary:** The solution for $x$ is $$\boxed{x = \pm \sqrt[r]{\frac{v}{2 - a}}}$$ This means $x$ is the $r$-th root of $\frac{v}{2 - a}$, with both positive and negative roots considered.