# What is the square root of 60?

##### 1 Answer

#### Explanation:

So we can simplify

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

It is not possible to simplify

Let

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

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

At each iteration,

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#