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The desired inequality is true for all $x \in (0, 1)$.

Using Bernoulli's inequality, 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.