True or False: $2^{2^{2011}} \text{ divides } 2^{2^{2012} }$ True or false: $$2^{2^{2011}} \text{ divides } 2^{2^{2012} }$$
Give your justifications. I don't know how to start this problem so far. 
But, I guessed like this,
$$2^{\underbrace{2\times 2\times\ldots\times 2}_{2011 times}}\text{ divides } 2^{\underbrace{2\times 2\times\ldots\times 2}_{2012 times}}$$
$$\implies \qquad \underbrace{2\times2\times\ldots\times 2}_{2^{2011}\;times}\text{ divides } \underbrace{2\times2\times\ldots\times 2}_{2^{2012}\;times}$$
Number of factors in the first product is less than that of second product. So defenitly,it will divide.
Kindly say is my reason is sound? If its ok, then give me alternate details about if we change the base .  $\left(\text{That is what about }a^{2^{2011}}\right.$ and $\left.b^{2^{2012}}\right)$ where $gcd(a,b)=d\neq 1$

True because $$2011 +1 = 2012 \Longrightarrow 2^{2012} = 2 \cdot 2^{2011} \Longrightarrow 2^{2^{2012}}= (2^{2^{2011}})^2$$
