Sendo A=(2sqrt (3))/(sqrt (6))+(4sqrt (2))/(3) e B=(5sqrt (75))/(sqrt (18))-2sqrt (6) determine o valor de B:A square

Para determinar o valor de \( B:A \), primeiro precisamos simplificar as expressões para \( A \) e \( B \).<br /><br />Para \( A \):<br />\[ A = \frac{2\sqrt{3}}{\sqrt{6}} + \frac{4\sqrt{2}}{3} \]<br /><br />Simplificando a primeira fração:<br />\[ \frac{2\sqrt{3}}{\sqrt{6}} = \frac{2\sqrt{3}}{\sqrt{2 \cdot 3}} = \frac{2\sqrt{3}}{\sqrt{2} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{2} \cdot \sqrt{3}} = \frac{2}{\sqrt{2}} = 2\sqrt{2} \]<br /><br />Então:<br />\[ A = 2\sqrt{2} + \frac{4\sqrt{2}}{3} \]<br /><br />Para \( B \):<br />\[ B = \frac{5\sqrt{75}}{\sqrt{18}} - 2\sqrt{6} \]<br /><br />Simplificando as raízes:<br />\[ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} \]<br />\[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]<br /><br />Então:<br />\[ \frac{5\sqrt{75}}{\sqrt{18}} = \frac{5 \cdot 5\sqrt{3}}{3\sqrt{2}} = \frac{25\sqrt{3}}{3\sqrt{2}} = \frac{25\sqrt{3}}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{25\sqrt{6}}{6} \]<br /><br />Então:<br />\[ B = \frac{25\sqrt{6}}{6} - 2\sqrt{6} \]<br /><br />Agora, para encontrar \( B:A \):<br />\[ B = \frac{25\sqrt{6}}{6} - 2\sqrt{6} = \frac{25\sqrt{6}}{6} - \frac{12\sqrt{6}}{6} = \frac{13\sqrt{6}}{6} \]<br /><br />Então:<br />\[ B:A = \frac{\frac{13\sqrt{6}}{6}}{2\sqrt{2} + \frac{4\sqrt{2}}{3}} \]<br /><br />Simplificando \( A \):<br />\[ A = 2\sqrt{2} + \frac{4\sqrt{2}}{3} = \frac{6\sqrt{2}}{3} + \frac{4\sqrt{2}}{3} = \frac{10\sqrt{2}}{3} \]<br /><br />Então:<br />\[ B:A = \frac{\frac{13\sqrt{6}}{6}}{\frac{10\sqrt{2}}{3}} = \frac{13\sqrt{6}}{6} \cdot \frac{3}{10\sqrt{2}} = \frac{13\sqrt{6} \cdot 3}{6 \cdot 10\sqrt{2}} = \frac{39\sqrt{6}}{60\sqrt{2}} = \frac{39\sqrt{3}}{60} = \frac{13\sqrt{3}}{20} \]<br /><br />Portanto, o valor de \( B:A \) é:<br />\[ \boxed{\frac{13\sqrt{3}}{20}} \]