Answer
$35\sqrt{2}+21\sqrt{5}$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{35}{5\sqrt{2}-3\sqrt{5}} \displaystyle \color{red}{ \cdot\frac{5\sqrt{2}+3\sqrt{5}}{5\sqrt{2}+3\sqrt{5}} }\qquad$ (rationalize)
$=\displaystyle \frac{35(5\sqrt{2}+3\sqrt{5})}{(5\sqrt{2})^{2}-(3\sqrt{5})^{2}} =\frac{35(5\sqrt{2}+3\sqrt{5})}{5^{2}\cdot 2-3^{2}\cdot 5}$
$ =\displaystyle \frac{35(5\sqrt{2}+3\sqrt{5})}{50-45}$
$=\displaystyle \frac{35(5\sqrt{2}+3\sqrt{5})}{5}$
$= 7(5\sqrt{2}+3\sqrt{5})$
$= 35\sqrt{2}+21\sqrt{5}$
