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#