How do you find the square root 28?
2 Answers
Unfortunately the only easy way to do this is by using a calculator to get
(this is the principal [nonnegative] square root)
Explanation:
If you must approximate
Note
#5^2 = 25 < 28color(white)("XXX")# so#color(white)("XXX")r> 5# #6^2= 36 > 28color(white)("XXX"# so#color(white)("XXX")r < 6#
taking a point between

taking a point between

Of course by now, you are probably wishing you had a calculator anyway to calculate the squared estimates...
As to calculating an approximation...
Explanation:
To calculate approximations of square roots of integers  especially if I'm trying to do it in my head or on a piece of paper  I like to find a sequence of rational approximations, stop at some point and long divide...
To do this I use a form of Newton Raphson method.
A standard formulation for finding the square root of a number
This is ok, but I prefer to split the numerator and denominator like this:
...to keep things in terms of integers rather than fractions for as long as possible:
Then our formulas for iterating become:
If iterating in this way happens to give a
Let's do this with
If we stopped at this stage we would have
If we stopped here, we would have
Giving
...accurate to
As you can see, this method converges quite fast, roughly doubling the number of significant digits on each iteration, but the numbers you are multiplying double in length on each iteration too.