Hey guys, i need to prove without using calculator that:

$\displaystyle 17 ^{14} >31^{11}$

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- Dec 17th 2010, 11:49 PMmat1990Which number is bigger
Hey guys, i need to prove without using calculator that:

$\displaystyle 17 ^{14} >31^{11}$ - Dec 18th 2010, 01:51 AMmelese
- Dec 21st 2010, 11:57 AMArchie Meade
Alternatively, if

$\displaystyle 16^{14}>32^{11}\Rightarrow\ 17^{14}>31^{11}$

$\displaystyle 16^{14}=16^3\left[16^{11}\right]=\left[2^4\right]^3\left[16^{11}\right]=2^{12}\left[16^{11}\right]$

$\displaystyle 32^{11}=2^{11}\left[16^{11}\right]$

$\displaystyle 2^{12}\left[16^{11}\right]>2^{11}\left[16^{11}\right]\Rightarrow\ 16^{14}>32^{11}\Rightarrow\ 17^{14}>31^{11}$