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:

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