What is the square root of 780?

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Answer 1 · 2015-10-23T21:25:23

$sqrt(780) = 2sqrt(195) ~~ 27.93$

$780 = 2^2 * 3 * 5 * 13$ has one square factor, so we can simplify $sqrt(780)$ using $sqrt(ab) = sqrt(a)sqrt(b)$ as follows:

$sqrt(780) = sqrt(2^2*195) = sqrt(2^2)sqrt(195) = 2sqrt(195)$

Now $195 = 14^2-1$ is of the form $n^2 - 1$ , so the continued fraction expansion of $sqrt(195)$ takes a simple form:

$sqrt(195) = [13;bar(1;26)] = 13 + 1/(1+1/(26+1/(1+1/(26+...))))$

We can approximate $sqrt(195)$ by truncating this continued fraction:

$sqrt(195) ~~ [13;1,26,1] = 13 + 1/(1+1/(26+1/1)) = 13+27/28 = 391/28 = 13.96dot(4)2815dot(7)$

So:

$sqrt(780) = 2sqrt(195) ~~ 391/14 = 27.9dot(2)8571dot(4) ~~ 27.93$