How do you simplify # sqrt19#?

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Answer 1 · 2016-05-16T19:52:44

#sqrt(19)# is already in simplest form.

#19# is a prime number, so has no square factors.

As a result it is not possible to simplify #sqrt(19)#, the positive irrational number whose square is #19#.

#sqrt(19) ~~ 4.35889894354#

The continued fraction expansion of #sqrt(19)# looks like this:

#sqrt(19) = [4; bar(2, 1, 3, 1, 2, 8)] = 4+1/(2+1/(1+1/(3+1/(1+1/(2+1/(8+1/(2+1/(1+...))))))))#

We can get an economical rational approximation for #sqrt(19)# by truncating just before the end of the repeating part:

#sqrt(19) ~~ [4; 2, 1, 3, 1, 2] = 4+1/(2+1/(1+1/(3+1/(1+1/2)))) = 170/39 = 4.bar(358974)#