# What is the simplest radical form of #sqrt115#?

##### 2 Answers

There is no simpler form

#### Explanation:

With radicals you try to factorize the argument, and see if there are any squares that can be 'taken out from under the root'.

Example:

In this case, no such luck:

#### Explanation:

The prime factorisation of

#115 = 5*23#

Since there are no square factors, it is not possible to simplify the square root. It is possible to express it as a product, but that does not count as simpler:

#sqrt(115) = sqrt(5)*sqrt(23)#

**Bonus**

In common with any irrational square root of a rational number,

#sqrt(115) = [10;bar(1,2,1,1,1,1,1,2,1,20)]#

#=10 + 1/(1+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(2+1/(1+1/(20+1/(1+...)))))))))))#

You can truncate the continued fraction expansion early to give rational approximations for

For example:

#sqrt(115) ~~ [10;1,2,1,1,1,1,1,2,1]#

#= 10 + 1/(1+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(2+1/1))))))))#

#=1126/105#

In fact, by truncating just before the end of the repeating section of the continued fraction, we have found the simplest rational approximation for

That is:

#115*105^2 = 1267875#

#1126^2 = 1267876#

only differ by

This makes