Answer
$$\frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{2}-\sqrt{5}}=\frac{9\sqrt{6}+3\sqrt{15}+3\sqrt{10}+5}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{2}-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{3}+\sqrt{5}}{3\sqrt{2}-\sqrt{5}}\frac{3\sqrt{2}+\sqrt{5}}{3\sqrt{2}+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{6}+3\sqrt{15}+3\sqrt{10}+5}{18+3\sqrt{10}-3\sqrt{10}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{6}+3\sqrt{15}+3\sqrt{10}+5}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{2} + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{3} + \sqrt{5}\right) } \cdot \left( 3 \sqrt{2} + \sqrt{5}\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{2}+\color{blue}{ 3 \sqrt{3}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot 3 \sqrt{2}+\color{blue}{ \sqrt{5}} \cdot \sqrt{5} = \\ = 9 \sqrt{6} + 3 \sqrt{15} + 3 \sqrt{10} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{2}- \sqrt{5}\right) } \cdot \left( 3 \sqrt{2} + \sqrt{5}\right) = \color{blue}{ 3 \sqrt{2}} \cdot 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot 3 \sqrt{2}\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 18 + 3 \sqrt{10}- 3 \sqrt{10}-5 $$ |
| ③ | Simplify numerator and denominator |
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