Which number is bigger

**mat1990** · December 17th 2010, 11:49 PM

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

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

**melese** · December 18th 2010, 01:51 AM

Originally Posted by mat1990

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

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

Notice that both 17 and 31 are "almost" powers of two, these are 16 and 32, respectively.

So $17^{14}>(2^4)^{14}=2^{56}$ , and $2^{55}=(2^5)^{11}>31^{11}$ . With this, you find the inequality....

**Archie Meade** · December 21st 2010, 11:57 AM

Originally Posted by mat1990

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

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

Alternatively, if

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

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

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

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

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