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:

$sqrt(10) ~~ 3.16227766016837933199$