# How do you find the exact square root of 41?

##### 1 Answer

#### Explanation:

We can use this expression to get rational *approximations* for

For example,

Alternatively, use a Newton Raphson type approach:

To find approximations for

#a_(i+1) = (a_i^2+n)/(2a_i)#

So if we started with

#a_1 = (a_0^2+n)/(2a_0) = (6^2+41)/12 = 77/12 = 6.41dot(6)#

#a_2 = (a_1^2+n)/(2a_1) = ((77/12)^2+41)/(2*(77/12))#

#= (77^2 + 41*12^2)/(2*77*12) = (5929+41*144)/1848#

#= (5929+5904)/1848 = 11833/1848 ~~ 6.40314#