How do you find the square root of 10?

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Answer 1 · 2016-06-26T20:10:10

#sqrt(10) ~~ 3.16227766016837933199# is not simplifiable.

#10 = 2xx5# has no square factors, so #sqrt(10)# is not simplifiable.

It is an irrational number a little greater than #3#.

In fact, since #10 = 3^2+1# is of the form #n^2+1#, #sqrt(10)# has a particularly simple continued fraction expansion:

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

We can truncate this continued fraction expansion to get rational approximations to #sqrt(10)#

For example:

#sqrt(10) ~~ [3;6] = 3+1/6 = 19/6 = 3.1bar(6)#

#sqrt(10) ~~ [3;6,6] = 3+1/(6+1/6) = 3+6/37 = 117/37 = 3.bar(162)#

Actually:

#sqrt(10) ~~ 3.16227766016837933199#