What is the square root of 60?

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Answer 1 · 2015-10-01T22:01:47

$sqrt(60) = 2 sqrt(15) ~~ 1921/248$

$60 = 2^2*3*5$ has a square factor $2^2$

So we can simplify $sqrt(60)$ using $sqrt(ab) = sqrt(a)sqrt(b)$ as follows:

$sqrt(60) = sqrt(2^2 * 15) = sqrt(2^2)sqrt(15) = 2sqrt(15)$

It is not possible to simplify $sqrt(15)$ further, but you can find rational approximations for it using a Newton Raphson type method.

Let $n = 15$ , $p_0 = 4$ , $q_0 = 1$ and iterate using the formulae:

$p_(i+1) = p_i^2 + n q_i^2$

$q_(i+1) = 2 p_i q_i$

At each iteration, $p_i/q_i$ is a rational approximation for $sqrt(n)$

So:

$p_1 = p_0^2 + n q_0^2 = 4^2 + 15*1^2 = 16+15 = 31$

$q_1 = 2 p_0 q_0 = 2*4*1 = 8$

Then:

$p_2 = p_1^2 + n q_1^2 = 31^2 + 15*8^2 = 961 + 960 = 1291$

$q_2 = 2 p_1 q_1 = 2 * 31 * 8 = 496$

We could go further to get a better approximation, but stop here to get:

$sqrt(15) ~~ 1291 / 496$

So

$sqrt(60) = 2sqrt(15) ~~ 2 * 1291 / 496 = 1291 / 248$