Prove the following, if true; otherwise, provide/derive a counterexample/contradiction:

1. If $\displaystyle n\in\mathbb{N}$ and $\displaystyle x\in\mathbb{R}^{+}$, then $\displaystyle x^n+x^{\frac{1}{n}}+n = 0$ has no real solutions for all $\displaystyle n \geqslant 2$.

2. If $\displaystyle n\in\mathbb{N}$ and $\displaystyle x \in\mathbb{R}^{+}$, then $\displaystyle x^n+x^{\frac{1}{n}}-n = 0$ has a single real solution for all $\displaystyle n \geqslant 1$.