How do you find the square root of 529?
Check for divisibility by a perfect square to simplify. You will find that
When we try to simplify a square root, we look for perfect square factors.
Do this by testing perfect squares until you get to a number whose square if greater that
So we test
Keep going . . .
Use a mixture of methods to find
There are quite a few different ways to find square roots.
Here's a bit of a mish-mash for this particular example...
Next, note that
We can approximate where
Hence a good first approximation for
We can use the Babylonian method to find a better approximation.
Given a positive number
#a_(i+1) = (a_i^2+n)/(2a_i)#
#a_1 = (a_0^2+n)/(2a_0) = (22^2+529)/(2*22) = (484+529)/44 = 1013/44 = 23.02bar(27)#
That looks suspiciously close to
#23^2 = 529#
Do a really rough estimate first using squares of multiples of
Now look at the last digit ...
There are only two numbers whose squares end with a
So the possibilities are
My first guess would therefore be
Multiplying confirms that