Prove this?

Question

17^14>31^11

I have the prove this without a calculator so I can't multiply.
I can use mathematical arguments only.

Answer 1

Edit: D'oh! Tan Z beat me in by 52 seconds! Anyway, after staring at this for while, I noticed that 17 is 1 more than 2^4 and 31 is 1 less than 2^5, so that inspired the following.

17^14 >
16^14 =
(2^4)^14 =
2^56 >
2^55 =
(2^5)^11 =
32^11 >
31^11

Answer 2

I'm not too sure about it, but here goes:

17^14
= (16+1)^14
= ((2^4)+1)^14

31^11
= (32 - 1)^11
= ((2^5) - 1)^11

Expanding both using binomial expansion.

((2^4)+1)^14 = 2^56 + 14(2^52) +..... (something, doesn't really matter)

((2^5)-1)^11 = 2^55 - 11(2^50) + ... (again, doesn't really matter)

The following terms get smaller and smaller. Anyway, The binomial expansion for 17^14 will always be '+'s, but for 31^11 it's alternating '+'s and '-'s.

Since 2^56 + 14(2^52) is already bigger than 2^55 - 11(2^50), we can conclude that 17^14 > 31^11.

That's how I'd prove it. Hope it helps!