Question

How do you simplify `sqrt10`?

Answer 1

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

Explanation:

`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)`