Formal Languages: Trouble understanding powers I've run into an argument with some peers about how to perform power operations on languages.
                               Say: A = {2,00}

At first myself and my peers were all in agreement that say for instance A^3 would output:
                       A = AAA = {(2,00),(2,00),(2,00)}

Likewise A^2:
                           A = AA = {(2,00),(2,00)}

However a fellow SO member said that this was not the case? ANd that say for instance A^3 would contain 8 elements?
Could anyone shed any light on who's right? And more importantly why? 
Thank you for your time 

If $A$ and $B$ are two languages, then $AB$ is the set of all strings $ab$ where $a\in A$ and $b\in B$. Thus $AB$ has $|A|\cdot|B|$ elements, where $|A|$ denotes the number of elements of $A$. Now $A^2=AA$ by definition, and $A^3=A^2A$ an so forth. So in your case, $A^2=\{22,200,002,0000\}$. Now I'll leave $A^3$ to you.
