The desired inequality is true for all $x \in (0, 1)$.
Using Bernoulli's inequality in the form $(1+u)^r\ge 1+ru$ for $r \ge 1$, $u\ge -1$, we have $$(1 - x)^{x^{-0.5}} \ge 1 - x \, x^{-0.5} = 1 - x^{1/2}.$$ Thus, we have $$x^{1/10}-(1-(1-x)^{x^{-0.5}}) \ge x^{1/10} - x^{1/2} \ge 0.$$
We are done.
