Difference of nth roots vs nth root of difference Show that for $n \ge 1, x>y \ge 0$ the following inequality holds: $x^{1\over n}-y^{\frac{1}{n}} \le (x-y)^{\frac{1}{n}}$. I have tried comparing the $n$-th powers of both sides of the equation using the binomial theorem but got bogged down in the process.

With $t:=(y/x)^{1/n}\in[0,1)$, you can rewrite

$$1-t\le(1-t^n)^{1/n}$$ or

$$(1-t)^n\le1-t^n.$$

This is obvious by

$$(1-t)^n\le1-t\le1-t^n.$$