What is the square root of 337?

1 Answer
May 13, 2016

Answer:

#sqrt(337) ~~ 18.35755975# is not simplifiable since #337# is prime.

Explanation:

#337# is prime - it has no positive factors apart from #1# and itself.

As a result, #sqrt(337)# is not simplifiable.

It is an irrational number which when squared (multiplied by itself) gives you #337#. Its value is approximately #18.35755975#.

Since it is irrational, its decimal representation neither terminates nor recurs.

It has a continued fraction expansion which does repeat, namely:

#sqrt(337) = [18;bar(2,1,3,1,11,2,4,1,3,3,1,4,2,11,1,3,1,2,36)]#

#=18+1/(2+1/(1+1/(3+1/(1+1/(11+1/(2+1/(4+1/(1+...))))))))#

To construct rational approximations for #sqrt(337)# you can truncate this continued fraction.

For example:

#sqrt(337) ~~ [18;2,1,3,1] = 18+1/(2+1/(1+1/(3+1/1))) = 257/14 ~~ 18.357#