5. \( \sqrt{\left(\frac{4}{9}+\frac{1}{2} \cdot \frac{7}{6}\right)+\frac{11}{9}}: \frac{6}{5}=x: \sqrt{\left(\frac{2}{5}+\frac{4}{3} \cdot \frac{3}{10}\right) \cdot \frac{1}{5}} \)

Let’s break down the given equation step by step! First, we simplify the expressions on both sides of the equation. On the left side, we have: \[ \frac{4}{9} + \frac{1}{2} \cdot \frac{7}{6} \] Calculating \( \frac{1}{2} \cdot \frac{7}{6} = \frac{7}{12} \). Now find a common denominator for \( \frac{4}{9} \) and \( \frac{7}{12} \), which is 36: \[ \frac{4}{9} = \frac{16}{36}, \quad \frac{7}{12} = \frac{21}{36} \] Adding them gives: \[ \frac{16}{36} + \frac{21}{36} = \frac{37}{36} \] Now add \( \frac{11}{9} \): Convert \( \frac{11}{9} \) to have a denominator of 36: \[ \frac{11}{9} = \frac{44}{36} \] So now we have \( \frac{37}{36} + \frac{44}{36} = \frac{81}{36} \). Now we can take the square root: \[ \sqrt{\frac{81}{36}} = \frac{9}{6} = \frac{3}{2} \] Now, we have: \[ \frac{3}{2} : \frac{6}{5} = x : \sqrt{\left(\frac{2}{5} + \frac{4}{3} \cdot \frac{3}{10}\right) \cdot \frac{1}{5}} \] Compute the right side: Calculating \( \frac{4}{3} \cdot \frac{3}{10} = \frac{4}{10} = \frac{2}{5} \): \[ \frac{2}{5} + \frac{2}{5} = \frac{4}{5} \] Now multiply by \( \frac{1}{5} \): \[ \left(\frac{4}{5}\right) \cdot \left(\frac{1}{5}\right) = \frac{4}{25} \] So we take the square root: \[ \sqrt{\frac{4}{25}} = \frac{2}{5} \] Now we have: \[ \frac{3}{2} : \frac{6}{5} = x : \frac{2}{5} \] Converting into fractions gives: \[ \frac{\frac{3}{2}}{\frac{6}{5}} = \frac{3}{2} \cdot \frac{5}{6} = \frac{15}{12} = \frac{5}{4} \] So now equate and solve for \( x \): \[ \frac{5}{4} = \frac{x}{\frac{2}{5}} \implies x = \frac{5}{4} \cdot \frac{2}{5} = \frac{2}{4} = \frac{1}{2} \] Thus, the final answer is: \[ \boxed{\frac{1}{2}} \]