How do you simplify the square root of 95?

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Answer 1 · 2018-03-17T00:35:07

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

is already in simplest form.

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:

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

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