Answer
$$\frac{7\sqrt{3}+2\sqrt{5}}{3\sqrt{5}-2\sqrt{3}}=\frac{25\sqrt{15}+72}{33}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{3}+2\sqrt{5}}{3\sqrt{5}-2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{3}+2\sqrt{5}}{3\sqrt{5}-2\sqrt{3}}\frac{3\sqrt{5}+2\sqrt{3}}{3\sqrt{5}+2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{21\sqrt{15}+42+30+4\sqrt{15}}{45+6\sqrt{15}-6\sqrt{15}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25\sqrt{15}+72}{33}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 \sqrt{3} + 2 \sqrt{5}\right) } \cdot \left( 3 \sqrt{5} + 2 \sqrt{3}\right) = \color{blue}{ 7 \sqrt{3}} \cdot 3 \sqrt{5}+\color{blue}{ 7 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{3} = \\ = 21 \sqrt{15} + 42 + 30 + 4 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}- 2 \sqrt{3}\right) } \cdot \left( 3 \sqrt{5} + 2 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot 3 \sqrt{5}\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3} = \\ = 45 + 6 \sqrt{15}- 6 \sqrt{15}-12 $$ |
| ③ | Simplify numerator and denominator |
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