What is a square root of a fraction that is rational?

If you are expected to calculate it without a calculator then the first step is to write the fraction as a ratio - a top heavy one if necessary. You will find that the numerator and denominator of the rational fraction are both perfect square numbers. Then the square root of the ratio is the ratio of the square roots.

For example, to find the square root of 11.11... recurring.

You know that, since it is a recurring decimal, the number is rational.

In ratio form, this number is 100/9 and so

sqrt(11.11...) = sqrt(100/9) = sqrt(100)/sqrt(9) = 10/3 = 3.33... recurring.