# Thread: Which number is bigger

1. ## Which number is bigger

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

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

2. Originally Posted by mat1990
Hey guys, i need to prove without using calculator that:

$\displaystyle 17 ^{14} >31^{11}$
Notice that both 17 and 31 are "almost" powers of two, these are 16 and 32, respectively.

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

3. Originally Posted by mat1990
Hey guys, i need to prove without using calculator that:

$\displaystyle 17 ^{14} >31^{11}$
Alternatively, if

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

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

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

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