# What is the square root of 780?

Oct 23, 2015

$\sqrt{780} = 2 \sqrt{195} \approx 27.93$

#### Explanation:

$780 = {2}^{2} \cdot 3 \cdot 5 \cdot 13$ has one square factor, so we can simplify $\sqrt{780}$ using $\sqrt{a b} = \sqrt{a} \sqrt{b}$ as follows:

$\sqrt{780} = \sqrt{{2}^{2} \cdot 195} = \sqrt{{2}^{2}} \sqrt{195} = 2 \sqrt{195}$

Now $195 = {14}^{2} - 1$ is of the form ${n}^{2} - 1$, so the continued fraction expansion of $\sqrt{195}$ takes a simple form:

sqrt(195) = [13;bar(1;26)] = 13 + 1/(1+1/(26+1/(1+1/(26+...))))

We can approximate $\sqrt{195}$ by truncating this continued fraction:

sqrt(195) ~~ [13;1,26,1] = 13 + 1/(1+1/(26+1/1)) = 13+27/28 = 391/28 = 13.96dot(4)2815dot(7)

So:

$\sqrt{780} = 2 \sqrt{195} \approx \frac{391}{14} = 27.9 \dot{2} 8571 \dot{4} \approx 27.93$