\[4\sqrt{6\frac{1}{4}} - 3\sqrt{1\frac{7}{9}} =\]
\[= 4\sqrt{\frac{25}{4}} - 3\sqrt{\frac{16}{9}} =\]
\[= 4 \cdot \frac{5}{2} - 3 \cdot \frac{4}{3} = 10 - 4 = 6.\]