2 Answers

square root(216) = 14.6969385

Here's one way to figure it out yourself:

Finding square roots by guess & check method

One simple way to find a decimal approximation to, say √2 is to make an initial guess, square the guess, and depending how close you got, improve your guess. Since this method involves squaring the guess (multiplying the number times itself), it actually uses the definition of square root, and so can be very helpful in teaching the concept of square root.

Example: what is √20 ?
Children first learn to find the easy square roots that are whole numbers, but quickly the question arises as to what are the square roots of all these other numbers. You can start out by noting that (dealing here only with the positive roots) since √16 = 4 and √25 = 5, then √20 should be between 4 and 5 somewhere.

Then is the time to make a guess, for example 4.5. Square that, and see if the result is over or under 20, and improve your guess based on that. Repeat the process until you have the desired accuracy (amount of decimals). It's that simple and can be a nice experiment for children.

Without a calculator:

14^2 = 196

15^2 = 225

square root of 216 lies between 14 and 15

14.9^2 = 222.01  >  too high

14.5^2 = 210.25  >  too low

GUESS

14.8^2 = 219.04    >     too high

14.7^2 = 216.09    >     too high

14.6^2 = 213.16    >     too low

14.69^2 = 215.7961  >  too low

14.695^2 = 215.943025  >  too low

14.699^2 = 216.060601  >  too high

14.698^2 = 216.031204  >  too high

14.697^2 = 216.001809  >  too high

14.696^2 = 215.972416  >  too low

14.6965^2 = 215.98711225  >  too low

14.6966^2 = 215.99005156  >  too low

14.6969^2 = 215.99886961  >  too low

14.69695^2 = 216.0003393025  >   high

14.69694^2 = 216.0000453636  >   high

14.69693^2 = 215.9997514249  >  close

14.696939^2 = 216.000015969721  >   high

14.696938^2 = 215.999986575844   >   close

I'm going to stop now.

The square root of 216 at six decimals is:

Answer: 14.696938

I rather use the calculator!