# How do you simplify sqrt205?

May 19, 2018

$\sqrt{205}$ is already in simplest form.

#### Explanation:

Note that:

$205 = 5 \cdot 41$

has no square factors larger than $1$. As a result, its square root cannot be simplified.

Using the algorithm described in https://socratic.org/s/aR4mTUXq we can find that the continued fraction expansion for $\sqrt{205}$ is:

sqrt(205) = [14;bar(3,6,1,4,1,6,3,28)]

$\textcolor{w h i t e}{\sqrt{205}} = 14 + \frac{1}{3 + \frac{1}{6 + \frac{1}{1 + \frac{1}{4 + \frac{1}{1 + \frac{1}{6 + \frac{1}{3 + \frac{1}{28 + \ldots}}}}}}}}$

From this we can find the smallest solution of Pell's equation for $205$:

${39689}^{2} = 205 \cdot {2772}^{2} + 1$

and related efficient rational approximation:

$\sqrt{205} \approx \frac{39689}{2772} \approx 14.31782107$