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

(i) $5<2.2+1.7+1.4<5^{1/2}+5^{1/3}+5^{1/4}$.
(ii) $8>2.9+2+1.7>8^{1/2}+8^{1/3}+8^{1/4}$.
(iii) $n\ge3\sqrt n>n^{1/2}+n^{1/3}+n^{1/4}$ because $n \ge 9$.
