Question

How do you find the square root of 404.41?

Answer 1

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`