# How do you simplify sqrt85?

Apr 30, 2016

$\sqrt{85}$ is not simplifiable.

#### Explanation:

The prime factorisation of $85$ is:

$85 = 5 \cdot 17$

This contains no square factors, so $\sqrt{85}$ is not simplifiable.

If you wish, you can factor it as:

$\sqrt{85} = \sqrt{5} \sqrt{17}$

but I don't think that counts as simplification.

The general idea is that if a radicand contains square factors then it can be simplified. For example:

$\sqrt{24} = \sqrt{{2}^{2} \cdot 6} = \sqrt{{2}^{2}} \sqrt{6} = 2 \sqrt{6}$

If it has no square factors then this kind of simplification is not possible.

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Random Bonus

Any square root is expressible as a repeating continued fraction. in our example:

sqrt(85) = [9;bar(4,1,1,4,18)]

$= 9 + \frac{1}{4 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4 + \frac{1}{18 + \frac{1}{4 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4 + \frac{1}{18 + \ldots}}}}}}}}}}$

You can truncate the continued fraction to give you rational approximations for the square root.

For example:

sqrt(85) ~~ [9;4] = 9+1/4 = 37/4 = 9.25

sqrt(85) ~~ [9;4,1,1,4] = 9+1/(4+1/(1+1/(1+1/(4)))) = 378/41 = 9.bar(21951)

Actually $\sqrt{85} \approx 9.219544457$