# How do you simplify sqrt65?

Mar 14, 2016

$65 = 5 \cdot 13$ has no square factors, so $\sqrt{65}$ is the simplest form.

#### Explanation:

If a radicand (the part under the root sign) of a square root has a square factor, then it can be simplified:

$\sqrt{{a}^{2} b} = \left\mid a \right\mid \sqrt{b}$

or if you know that $a \ge 0$, more simply:

$\sqrt{{a}^{2} b} = a \sqrt{b}$

For example, $\sqrt{24} = \sqrt{{2}^{2} \cdot 6} = 2 \sqrt{6}$

In our example, we find $65 = 5 \cdot 13$ has no square factors, so cannot be simplified in this way.

If you like, you can reexpress it:

$\sqrt{65} = \sqrt{5} \sqrt{13}$

but that is not (as far as I know) considered 'simpler'.

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Note that $\sqrt{65}$ is an irrational number. That is, it cannot be expressed as a fraction $\frac{p}{q}$ for integers $p$ and $q$. As a result, its decimal expansion does not terminate or recur.

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Bonus

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

is of the form ${n}^{2} + 1$ with $n = 8$.

As a result, the square root can be expressed as a very simple continued fraction ...

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

You can use this to give you good approximations for $\sqrt{65}$, by truncating the continued fraction after a few terms.

For example,

[8; 16] = 8+1/16 = 129/16 = 8.0625

[8; 16, 16] = 8+1/(16+1/16) = 8+16/257 = 2072/257 ~~ 8.0622568

Actually $\sqrt{65}$ is closer to $8.06225774829854965236$