How do you simplify the square root of 95?

1 Answer
Mar 17, 2018

Answer:

#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#