# What is sqrt65 in simplified radical form?

Jun 19, 2016

$\sqrt{65}$ cannot be simplified.

#### Explanation:

The prime factorisation of $65$ is:

$65 = 5 \cdot 13$

Since this has no square factors $\sqrt{65}$ cannot be simplified.

Bonus

$65 = 64 + 1 = {8}^{2} + 1$

is in the form ${n}^{2} + 1$.

The square roots of such numbers have a simple form of continued fraction expansion:

sqrt(n^2+1) = [n;bar(2n)] = n+1/(2n+1/(2n+1/(2n+1/(2n+1/(2n+1/(2n+...))))))

So in our example:

sqrt(65) = [8;bar(16)] = 8+1/(16+1/(16+1/(16+1/(16+1/(16+1/(16+...))))))

You can get rational approximations of $\sqrt{65}$ to any desired accuracy by truncating the continued fraction expansion early.

For example:

sqrt(65) ~~ [8;16] = 8+1/16 = 8.0625

sqrt(65) ~~ [8;16,16] = 8+1/(16+1/16) = 8+16/257 ~~ 8.0622568

The actual value is more like:

$\sqrt{65} \approx 8.0622577483$