# How do you simplify sqrt(1414)?

Jun 3, 2017

$\sqrt{1414}$ is already in simplest form.

#### Explanation:

The prime factorisation of $1414$ is:

$1414 = 2 \cdot 7 \cdot 101$

This contains no square factors, so the square root is already in simplest form.

$\textcolor{w h i t e}{}$
Notes

Should the $1414$ in the question have been $1444$?

If it was, then we would find:

$1444 = 2 \cdot 2 \cdot 19 \cdot 19 = {38}^{2}$

So:

$\sqrt{1444} = 38$

We also find that:

${37}^{2} = 1369 < 1414 < 1444 = {38}^{2}$

So $\sqrt{1414}$ is an irrational number somewhere between $37$ and $38$.

If you would like a rational approximation, we can start by linearly interpolating between $37$ and $38$, finding:

$\sqrt{1414} \approx 37 + \frac{1414 - 1369}{1444 - 1369} = 37 + \frac{45}{75} = 37 + \frac{3}{5} = \frac{188}{5}$

This will be slightly less than $\sqrt{1414}$.

We find:

$\left(\frac{188}{5} ^ 2\right) = \frac{35344}{25} = \frac{35350}{25} - \frac{6}{25} = 1414 - \frac{6}{25}$

That's not bad, but if we want greater accuracy, we can use a generalised continued fraction based on this approximation.

In general we have:

$\sqrt{{a}^{2} + b} = a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}}$

Putting $a = \frac{188}{5}$ and $b = \frac{6}{25}$ we get:

$\sqrt{1414} = \frac{188}{5} + \frac{\frac{6}{25}}{\frac{376}{5} + \frac{\frac{6}{25}}{\frac{376}{5} + \frac{\frac{6}{25}}{\frac{376}{5} + \ldots}}}$

You can terminate this continued fraction to get rational approximations, such as:

$\sqrt{1414} \approx \frac{188}{5} + \frac{\frac{6}{25}}{\frac{376}{5} + \frac{\frac{6}{25}}{\frac{376}{5}}} = \frac{13291036}{353455} \approx 37.603191353921$

A calculator tells me that $\sqrt{1414}$ is a little closer to:

$37.60319135392633134161$