How do you find the square root of 12025?

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Answer 1 · 2017-02-05T10:34:50

#sqrt(12025) = 5sqrt(481) ~~ 109.658561#

Note that:

#12025 = 5^2*481 = 5^2*13*37#

So:

#sqrt(12025) = sqrt(5^2*481) = 5sqrt(481)#

We can find rational approximations for #sqrt(481)# using a form of the Babylonian method:

Given a rational approximation #p/q# to #sqrt(n)#, we can find a better approximation by calculating:

#(p^2+nq^2)/(2pq)#

In our example, #481# is quite close to #484 = 22^2#, so use #22/1# as our first approximation.

The next approximation would be:

#(22^2+481*1^2)/(2*22*1) = (484+481)/44 = 965/44#

For more accuracy, iterate again:

#(965^2+481*44^2)/(2*965*44) = (931225+931216)/84920 = 1862441/84920 ~~ 21.9317122#

So:

#sqrt(12025) ~~ 5*21.9317122 = 109.658561#