How do you find the square root of 730?

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Answer 1 · 2016-09-16T22:58:17

$sqrt(730) ~~ 78813/2917 ~~ 27.01851217$

$730 = 2*5*73$ has no square factors, so the square root cannot be simplified.

It is an irrational number a little greater than $27 = sqrt(729)$

Since $730 = 27^2+1$ is in the form $n^2+1$ it has a simple form of continued fraction:

$sqrt(730) = [27;bar(54)] = 27+1/(54+1/(54+1/(54+1/(54+...))))$

We can truncate this continued fraction early to get rational approximations for $sqrt(730)$ .

For example:

$sqrt(730) ~~ [27;54] = 27+1/54 = 1459/54 = 27.0bar(185)$

or more accurately:

$sqrt(730) ~~ [27;54,54] = 27+1/(54+1/54) = 78813/2917 ~~ 27.01851217$