Show that equation
accepts a positive single resolution no rational.
As this is a polynomial of odd degree, it has at least one real solution. Using calculus (we are in the right section for using calculus here ) shows that the function $\displaystyle f(x)=x^5+x-10$ is strictly increasing for all $\displaystyle x$ and so the equation has at most one real solution.
So there is exactly one real solution. It remains to show that this solutiosn is positive and not rational. The former is easy: $\displaystyle f(1)=-8<0$ and $\displaystyle f(2)=24>0$ so the solution lies between $\displaystyle x=1$ and $\displaystyle x=2.$ I leave the rest to you.