How do you simplify #sqrt10#?

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Answer 1 · 2016-02-17T00:09:34

It is not possible to simplify #sqrt(10)#, but we can find rational approximations quite easily...

#10 = 2*5# has no square factors, so #sqrt(10)# has no simpler form.

It is an irrational number, that is it not expressible as #p/q# for integers #p# and #q#. Neither will its decimal expansion repeat or terminate.

#10 = 3^2 + 1#, hence #sqrt(10)# is a little more than #3#.

In fact, it can be expressed as a continued fraction:

#sqrt(10) = [3;bar(6)] = 3 + 1/(6+1/(6+1/(6+1/(6+...))))#

We can get rational approximations for #sqrt(10)# by terminating this continued fraction.

For example:

#sqrt(10) ~~ 3 + 1/(6+1/6) = 117/37 ~~ 3.bar(1)6bar(2)#