Answer
$$\frac{2\sqrt{5}+\sqrt{3}}{3\sqrt{5}-\sqrt{3}}=\frac{33+5\sqrt{15}}{42}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}+\sqrt{3}}{3\sqrt{5}-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}+\sqrt{3}}{3\sqrt{5}-\sqrt{3}}\frac{3\sqrt{5}+\sqrt{3}}{3\sqrt{5}+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30+2\sqrt{15}+3\sqrt{15}+3}{45+3\sqrt{15}-3\sqrt{15}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{33+5\sqrt{15}}{42}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{5} + \sqrt{3}\right) } \cdot \left( 3 \sqrt{5} + \sqrt{3}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot 3 \sqrt{5}+\color{blue}{ \sqrt{3}} \cdot \sqrt{3} = \\ = 30 + 2 \sqrt{15} + 3 \sqrt{15} + 3 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}- \sqrt{3}\right) } \cdot \left( 3 \sqrt{5} + \sqrt{3}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot 3 \sqrt{5}\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 45 + 3 \sqrt{15}- 3 \sqrt{15}-3 $$ |
| ③ | Simplify numerator and denominator |
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