2^10000= 10^(lg2*10000)= 10^3010.2999566398119521373889472449
If your teacher doesn't like that answer

pick up the decimals of lg2*10000=3010.x

so you get 10^0.x = 1.9950631168807583848837421626836

Thus you get 1.9950631168807583848837421626836*10^3010

Haven't anyone here learned to count with logaritmes?
This is a better method, because it will work with any number, whereas my suggestion only works with powers of 2.

And now I feel the effects of not having taken a math class in the last five years. When I was taking AP Calc I was on top of the world; now the good Duke's post is complete gibberish to me.
And I have not done a maths course for 23 years. Is there a smug smiley?