# How do you find the square root of 730?

Sep 16, 2016

$\sqrt{730} \approx \frac{78813}{2917} \approx 27.01851217$

#### Explanation:

$730 = 2 \cdot 5 \cdot 73$ has no square factors, so the square root cannot be simplified.

It is an irrational number a little greater than $27 = \sqrt{729}$

Since $730 = {27}^{2} + 1$ is in the form ${n}^{2} + 1$ it has a simple form of continued fraction:

sqrt(730) = [27;bar(54)] = 27+1/(54+1/(54+1/(54+1/(54+...))))

We can truncate this continued fraction early to get rational approximations for $\sqrt{730}$.

For example:

sqrt(730) ~~ [27;54] = 27+1/54 = 1459/54 = 27.0bar(185)

or more accurately:

sqrt(730) ~~ [27;54,54] = 27+1/(54+1/54) = 78813/2917 ~~ 27.01851217