When $2^{2012}$ is multiplied by $5^{2013}$, what is the sum of all the digits in the product?
I know that if the base is the same, I can add the powers, but the base is not the same, nothing matches an formula. I don't get it, I mean the only way I think I can find the answer is if I find out the value of $2^{2012} \cdot 5^{2013}$. I tried to mannually do the equation and got 5 as the anser, but in a test, I wouldnt have enough time to do the actually thing, so is there an easier way to do it?
Hint: you've told us that there's a rule for when bases are the same, $x^ax^b=x^{a+b}$.
What's the rule for when the exponents are the same?: $a^xb^x=(ab)^x$?
Then, notice that $2^{2012}$ and $5^{2013}$ do not have exactly the same exponents, but they very nearly do...
You certainly will be able to find the value of $2^{2012}\times 5^{2013}$, even if this value is too large to actually write out.