Chapter 9 - Roots and Radicals - 9.3 - More on Simplifying Radicals - Problem Set 9.3 - Page 412: 39

$\dfrac{\sqrt{37}}{3}$

Recall, we are not allowed to have radicals in the denominator of a fraction. Using the properties of radicals, the given expression, $ \sqrt{4\dfrac{1}{9}} ,$ simplifies to \begin{array}{l}\require{cancel} \sqrt{\dfrac{37}{9}} \\\\= \dfrac{\sqrt{37}}{\sqrt{9}} \\\\= \dfrac{\sqrt{37}}{\sqrt{(3)^2}} \\\\= \dfrac{\sqrt{37}}{3} .\end{array}