What is $57^{46}$ divided by 17? $$\frac{57^{46}}{17}$$

What is best and quickest way to evaluate an approximate answer to this by hand?
It really depends what you mean by "approximate answer". Since you tagged arithmetic...

Notice 57 = 3 . 19, you can say that $57^{46} / 17 = 3 \cdot 57^{45} \cdot \frac{19}{17}$. So if you know what $57^{45}$ is, you can approximate to within 2% if you multiply that by 3.3. 

Now, compute by hand $57^5 = 601692057$, which is less than 0.3% off from $6\cdot 10^8$. Observe that $57^{45} = (57^5)^9$. Using the binomial theorem you see that you can approximate that by $(6\cdot 10^8)^9 = 10 077 696 \cdot 10^{72}$ to within 3%. Rounding off the lower digits won't matter much in the error, so you have $57^{45} \sim 10^{79}$ to 3%. 

So putting it all together you have that to within 5%, 

$$ 57^{46} \sim 3.3 \cdot 10^{79} $$
The result is exactly:
$
34657277165715299429841134098673896005946880584360167911317357675111364398875720+\frac{9}{17}
$