Proving AM GM Inequality & Solving for Integers | Homework Help

[The legend]

Homework Statement

Prove

a)5 < 5^1/2 + 5^1/3 + 5^1/4

b) n > n^1/2 + n^1/3 + n^1/4 for all ints n>8

Homework Equations

The Attempt at a Solution

i tried the AM GM inequality
and found

5^1/2 + 5^1/3 + 5^1/4 > 3(5^13/36)

what further can i do?
can anyone please help me out??

 

[Mark44]

Homework Statement

Prove

a)5 < 5^1/2 + 5^1/3 + 5^1/4

b) n > n^1/2 + n^1/3 + n^1/4 for all ints n>8

Homework Equations

The Attempt at a Solution

i tried the AM GM inequality
and found

5^1/2 + 5^1/3 + 5^1/4 > 3(5^13/36)

> 3 * 5^1/3 = (27 * 5)^1/3

Can you show that the last expression is > 5?

what further can i do?
can anyone please help me out??

 

[The legend]

Thanks
so i got that part and showed the expression > 5.

I could prove the 2nd part by this method too!

Thanks a lot!