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

HINT: For (ii) we have to prove that
$$6>2\sqrt{2}+\sqrt[4]{8}$$
This can be written as
$$6-2\sqrt{2}>\sqrt[4]{8}$$
Raise to the power $4$
we have
$$3080>2112\sqrt{2}$$
Squaring we get
$$565312>0$$
For (i) we get by $$AM-GM$$
$$\frac{5^{1/2}+5^{1/3}+5^{1/4}}{3}\geq \sqrt[3]{5^{13/12}}$$ so we have to Show that
$$3\sqrt[3]{5^{13/12}}>5$$ this is equivalent to $$27^{12}>5^{23}$$ and this is 
$$150094635296999121>11920928955078125$$
