# How do you find the square root of 404.41?

Sep 21, 2015

Use a Newton Raphson type method to find:

$\sqrt{404.41} \approx 20.10994779$

#### 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 $\sqrt{404.41} = \sqrt{\frac{40441}{100}} = \frac{\sqrt{40441}}{\sqrt{100}} = \frac{\sqrt{40441}}{10}$

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:

$40441 = 37 \cdot 1093$

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:

${p}_{1} = {p}_{0}^{2} + n {q}_{0}^{2} = {200}^{2} + 40441 \cdot {1}^{2} = 80441$
${q}_{1} = 2 {p}_{0} {q}_{0} = 2 \cdot 200 \cdot 1 = 400$

${p}_{2} = {80441}^{2} + 40441 \cdot {400}^{2} = 12941314481$
${q}_{2} = 2 \cdot 80441 \cdot 400 = 64352800$

This gives an approximation:

$\sqrt{40441} \approx \frac{12941314481}{64352800} \approx 201.09947789$

Hence $\sqrt{404.41} \approx 20.10994779$

Actually $\sqrt{40441} \approx 201.09947787$