When I approach something like this to simplify, the first thing I check for is: does the fraction simplify to something else? Here, the fraction in question is 40/169, which doesn't simplify down any further, so that first check isn't really applicable in this case.
Since I can't simplify the fraction, I'll approach the square root of the fraction as the square root of the numerator divided by the square root of the denominator. Therefore, we are now looking at two sub-problems: simplifying the square root of 40 and simplifying the square root of 169.
Let's start off with the easy one - for memorizing your squares comes in handy here, since 169 is equal to 13 squared. If that isn't memorized, no fret, because with trial and error and a little patience, you'll still get the same answer. So, we know that the final answer has 13 as the denominator.
Now let's take a look at the square root of 40. The most comprehensive advice I have for how to simplify this is to break down the number (in this case 40) to its prime factorization, which would be 2*2*2*5. Since there is a pair of 2's, you can take them out from under the square root, leaving 2*5 under the root symbol since they don't make a pair. Like the answer indicated above, you now have 2 root 5 in the numerator.
Putting the numerator and denominator together gives: (2 root 10) / 13.
`sqrt(40/169) = (sqrt(4)*sqrt(10))/sqrt(169)`