How do you find the square root of 404.41?

1 Answer
Sep 21, 2015

Answer:

Use a Newton Raphson type method to find:

#sqrt(404.41) ~~ 20.10994779#

Explanation:

#404.41 = 20^2 + 2.1^2#, so you might think there's a nice expression for #sqrt(404.41)#, but not so.

We can say #sqrt(404.41) = sqrt(40441/100) = sqrt(40441)/sqrt(100) = sqrt(40441)/10#

So the problem reduces to finding the square root of the whole number #40441# then dividing by #10#.

What's the prime factorisation of #40441#?

Trying each prime in turn, we eventually find:

#40441 = 37 * 1093#

So #40441# has no square factors and the square root cannot be simplified.

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 #200# as follows:

#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 #sqrt(404.41) ~~ 20.10994779#

Actually #sqrt(40441) ~~ 201.09947787#