Darlene,

1. Usually in a timed test without a calculator you are allowed
   to leave the answer as √42563 or whatever. Check if this is the case in your class.
   
   
2. 42563 is not a perfect square (it ends in 3; no perfect square does) If you are told that a number of that approximate size has a natural number square root (eg, 42436) you can work it out fairly quickly. Start by breaking the digits into 2-digit "periods":
   
         4 24 36
   
   The square root will be between 200 and 300. Now you need to experiment a little. 210 x 210 = 44100 which is too big so the square root is between 200 and 210. And your number ends in 6; of the numbers between 1 and 10 only 4 and 6 have squares ending in 6, so test 204 and 206. The second one works.
   
   
3. If you do not have reason to suppose the number to be a perfect square, you can use the same method but you will have to settle for an approximate answer; if a square root of a natural number is not a natural number its decimal will never terminate or repeat. So for 42563
   you find (as above) that the root is between 200 and 210. 42563 is about halfway between 40000 and 44100 so try 205. Its square is 42025, a bit too small. 206 squared is 42436, 207 squared is 42849; so your answer is 206 and a bit. [I am not using a calculator for any of this!]
   
   

In this small an interval the graph of the square root function is very close to linear, so a proportional interpolation will give you a lot more precision. The idea is that the square root divides the interval [206,207] in very nearly the same proportions that 42563 divides the interval [42436,42849].

42563 - 42436 = 127
42849 - 42436 = 413

Check using Google - true value is 206.308022... We're out by one part in a hundred thousand. Close enough for rock'n'roll.

Recap: get first digit and place value by pairing. Then use one or two rounds of trial squaring to get the next two significant figures. Then use linear interpolation to get another three. This value will be good enough for all practical purposes.

You can also streamline this by doing ONE round of trial squaring and linear interpolation. Don't keep too many decimal places in the division as they won't be right beyond the second.

200² = 40000
210² = 44100

42563 - 40000 = 2563
44100 - 40000 = 4100

2562/4100 = 0.62... * (210-200) = 6.2...

estimated value 206.2; not too bad.

Good Hunting!
RD