Hello,

If the number in question is small enough, as in your case, it is simple enough to find the square root by trial and error.  We are looking for a number which, when multiplied by itself, gives the square number we're supplied with.

For example, take 169.  Since 10 x 10 = 100, we know we can start above 10.  So try...

11 x 11 = 121
12 x 12 = 144
13 x 13 = 169 and there we have it! 

The square root of 169 is 13.  These multiplications are relatively easy assuming you have a certain level of skill, they can be done in your head or on a napkin with a pen.

Now for 217, we know we have to start above 13, so try...

14 x 14 = 196
15 x 15 = 225

But now we're above the number we're trying to get, 217.  That means that the square root of 217 has to be between 14 and 15.

If we want to get closer to the answer, we could use a slightly more complicated process that involves guessing the answer, checking how close we are, and then adjusting our guess based on this.  The process would work like the following:

Suppose we guess that the square root of 217 is 14.5.  If we divide 217 by 14.5, we get about 14.9655.  This means that our guess of 14.5 was too low.  So if we then take the average of 14.5 and 14.9655 we'll get a number closer to the true square root of 217.  The average of 14.5 and 14.9655 is about 14.7328.  Then if we divide 217 by this new number, 14.7328, we get 14.7291.  That means that 14.7328 is actually a little bit greater than the true square root of 217. 

We can go on this way homing in on the square root, but the calculations (especially if you don't have a calculator) can get cumbersome.  From the steps that we've taken so far, it would seem safe to assume that the square root of 217 is approximately 14.7, but this is only to one decimal place.  It depends on how much precision you want in your answer how far you should go down this road.  This technique of guessing, checking and then guessing again based on the average is one of the many techniques known as iterative numerical approximation.

If it seems like this in an unsatisfying answer, you may be interested to know that the square root of almost any number is an "irrational" number.  An irrational number is a number that has a never-ending, never-repeating number of decimals!  It is called "irrational" because it cannot be expressed as a ratio of two numbers.  Well known irrational numbers are π , e, and the square root of any positive whole number which is not a "perfect square" (for example, the square root of 2, 3, 5, 6, 7, 8, 10, 11, 12, 13 etc.).  This means that, no matter how far we go in approximating a number, we will never get to the "true" square root, it will always be an approximation.  It is most useful to set ourselves a limit on precision and then approximate until we are below that limit.  This is essentially what your calculator does when it computes the square root of a number.

I hope this helps you!
Gabe