Answer:
[tex]\frac{(5\, \sqrt{3} +3\,\sqrt{5})^2 }{30 }[/tex]
Step-by-step explanation:
We recall that in order to rationalize this type of denominator, we can use the multiplication (in numerator and denominator) by the denominator's conjugate:
[tex]\frac{5\, \sqrt{3} +3\,\sqrt{5} }{5\, \sqrt{3} -3\,\sqrt{5} } =\frac{5\, \sqrt{3} +3\,\sqrt{5} }{5\, \sqrt{3} -3\,\sqrt{5} }*\frac{5\, \sqrt{3} +3\,\sqrt{5} }{5\, \sqrt{3} +3\,\sqrt{5} }=\\=\frac{(5\, \sqrt{3} +3\,\sqrt{5})^2 }{(5\, \sqrt{3} )^2-(3\,\sqrt{5})^2 }=\frac{(5\, \sqrt{3} +3\,\sqrt{5})^2 }{(25*3)-(9*5) }=\\=\frac{(5\, \sqrt{3} +3\,\sqrt{5})^2 }{30 }[/tex]
