a way to get close to big square roots in your head

First take your number, say 25347 and round that a bit to
2.5*10^4
note that the 10^4 is an even power of 10 so use this formula:
.8+(1/3)*x-(1/100)*x^2
so .8 + (1/3)2.5 = 1.63 and subtract .0625 to get 1.57
for every even power of 10 above 0 move a decimal place to the right, so since 25347 was *10^4 move 2 decimal places over. That gives you 157 which is close to the square root of 25347 (it is actually 159.2)

For odd powers such as 253478
use this formula:
2.4+1.1*x - (1/30)*x^2
so start the same round the number to
2.5 *10^5
2.4 + 1.1*2.5 = 2.4 + 2.75 = 5.15 and then do 1/3 of 2.5 .83 and shift to .083, 5.15-.083 = 5.067
then shift the decimal to the right for every odd power of 10 over 1, 506 the answer is actually around 503.

So with a little practice one can get pretty good at getting close to square roots in one's head.

Why it works:

Here's a graph of the function I listed vs. the square root from 1 to 10

The shape of the graph repeats for 100-1000 with the y axis multiplied by 10

Every even power of 10 such as 4.5*10^4 looks just like 4.5 but with the axis multiplied by a 100. 

The other function works the same way.