<s>[INST] Inequality in number theory Prove that:
(i)$5<5^{\frac{1}{2}}+5^{\frac{1}{3}}+5^{\frac{1}{4}}$
(ii)$8>8^{\frac{1}{2}}+8^{\frac{1}{3}}+8^{\frac{1}{4}}$
(iii)$n>n^{\frac{1}{2}}+n^{\frac{1}{3}}+n^{\frac{1}{4}}$ for all integer $n\geq9$
Can raising both sides to exponent $12$ help
 [/INST] The last one is equivalent to showing 
$$1 > \frac{1}{n^{\frac{1}{2}}}+\frac{1}{n^{\frac{2}{3}}}+\frac{1}{n^{\frac{3}{4}}}$$
and 
$$n\geq 9 \Rightarrow \frac{1}{3}\geq \frac{1}{n^{\frac{1}{2}}}$$
$$n^2\geq 9^2 > 3^3 \Rightarrow \frac{1}{3}> \frac{1}{n^{\frac{2}{3}}}$$
$$n^3\geq 9^3 > 3^4 \Rightarrow \frac{1}{3}> \frac{1}{n^{\frac{3}{4}}}$$
and the result follows.
</s>