# How do you find the square root of 15?

Jun 26, 2016

$\sqrt{15}$ is not simplifiable.

We can find rational approximations $\frac{31}{8}$, $\frac{244}{63}$

#### Explanation:

$15 = 3 \times 5$ has no square factors, so $\sqrt{15}$ cannot be simplified.

It is not expressible as a rational number. It is an irrational number a little less than $4$.

Since $15 = {4}^{2} - 1$ is of the form ${n}^{2} - 1$, $\sqrt{15}$ has a fairly simple continued fraction expansion:

sqrt(15) = [3;bar(1,6)] = 3+1/(1+1/(6+1/(1+1/(6+1/(1+1/(6+1/(1+...)))))))

We can truncate this continued fraction expansion early to get rational approximations to $\sqrt{15}$.

For example:

sqrt(15) ~~ [3;1,6,1] = 3+1/(1+1/(6+1/1)) = 3+1/(1+1/7) = 3+7/8 = 31/8 = 3.875

sqrt(15) ~~ [3;1,6,1,6,1] = 3+1/(1+1/(6+1/(1+1/(6+1/1)))) = 3+1/(1+1/(6+1/(1+1/7)))

$= 3 + \frac{1}{1 + \frac{1}{6 + \frac{7}{8}}} = 3 + \frac{1}{1 + \frac{8}{55}} = 3 + \frac{55}{63} = \frac{244}{63} = 3. \overline{873015}$

Actually:

$\sqrt{15} \approx 3.87298334620741688517$