# Thread: whitch is larger

1. ## whitch is larger

$\displaystyle 2^{3^{2^{3^{2^3}}}}$ or$\displaystyle 3^{2^{3^{2^{3^2}}}}$

2. Recall that:

$\displaystyle (x^a)^b = x^{ab}$

So simply multiply all of the exponents out to simplify your expression:

$\displaystyle 2^{3^{2^{3^{2^3}}}} = 2^{108}$

and:

$\displaystyle 3^{2^{3^{2^{3^2}}}} = 3^{72}$

Now simply punch the two into your calculator.

3. Hello,

I think that the powers are not with the parenthesis.

According to what he really wrote, this means that $\displaystyle 2^{3^2} = 2^9$, not $\displaystyle (2^3)^2 = 2^6$. It should be a bit more complicated

4. $\displaystyle 2^{3^{2^3}}=2^{3^8}=2^{6561}$

$\displaystyle 3^{2^{3^2}}=3^{2^9}=3^{512}$

$\displaystyle 2^{6561}$ is a very much bigger number than $\displaystyle 3^{512}$, and since $\displaystyle \ln{3}>\ln{2}$, therfore $\displaystyle 2^{6561}\ln{3}$ is a very much bigger number than $\displaystyle 3^{512}\ln{2}$. On the other hand, $\displaystyle \ln{(\ln{2})}<\ln{(\ln{3})}$ but the difference is minuscule in comparison. Therefore, I think we can conclude that

$\displaystyle 2^{6561}\ln{3}+\ln{(\ln{2})}\ >\ 3^{512}\ln{2}+\ln{(\ln{3})}$

that is to say,

$\displaystyle 2^{3^{2^{3^{2^3}}}}\ >\ 3^{2^{3^{2^{3^2}}}}$