# How do you find the square root of 404.41?

##### 1 Answer

Use a Newton Raphson type method to find:

#### Explanation:

We can say

So the problem reduces to finding the square root of the whole number

What's the prime factorisation of

Trying each prime in turn, we eventually find:

So

To find a good approximation:

See my answer to: How do you find the square root 28?

Use a Newton Raphson type method with an initial approximation of

#n = 40441#

#p_0 = 200#

#q_0 = 1#

Iteration step:

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

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

So:

#p_1 = p_0^2 + n q_0^2 = 200^2 + 40441 * 1^2 = 80441#

#q_1 = 2 p_0 q_0 = 2 * 200 * 1 = 400#

#p_2 = 80441^2 + 40441 * 400^2 = 12941314481#

#q_2 = 2 * 80441 * 400 = 64352800#

This gives an approximation:

#sqrt(40441) ~~ 12941314481 / 64352800 ~~ 201.09947789#

Hence

Actually