Question

How do you simplify the square root of 95?

Answer 1

`sqrt(95) ~~ 18495361/1897584 ~~ 9.746794344809`

is already in simplest form.

Explanation:

The prime factorisation of `95` is:

`95 = 5 * 19`

Since this contains no square factors, `sqrt(95)` is already in simplest form. There are no factors that can be moved outside the radical.

As a continued fraction, we find:

`sqrt(95) = [9;bar(1,2,1,18)] = 9+1/(1+1/(2+1/(1+1/(18+1/(1+1/(2+1/(1+1/(18+...))))))))`

Hence an efficient rational approximation for `sqrt(95)` is:

`[9;1,2,1] = 9+1/(1+1/(2+1/1)) = 39/4`

For a fun way to find better approximations to `sqrt(95)`, consider the quadratic with zeros `39+4sqrt(95)` and `39-4sqrt(95)`:

`(x-39-4sqrt(39))(x-39+4sqrt(95)) = (x-39)^2-1520`

`color(white)((x-39-4sqrt(39))(x-39+4sqrt(95))) = x^2-78x+1`

Based on this, define a sequence recursively by:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 78a_(n+1)-a_n) :}`

The first few terms of this sequence are:

`0, 1, 78, 6083, 474396, 36996805`

The ratio between successive terms will converge rapidly to `39+4sqrt(95)`.

Hence we find:

`sqrt(95) ~~ 1/4(36996805/474396 - 39) = 1/4(18495361/474396) = 18495361/1897584`

`color(white)(sqrt(95)) ~~ 9.746794344809`