\[8\sqrt{2\frac{1}{4}} - 3\sqrt{5\frac{4}{9}} =\]
\[= 8\sqrt{\frac{9}{4}} - 3\sqrt{\frac{49}{8}} = 8 \cdot \frac{3}{2} - 3 \cdot \frac{7}{3} =\]
\[= 12 - 7 = 5.\]