Answer
$$\frac{3\sqrt{2}+7\sqrt{5}}{2\sqrt{3}+6\sqrt{5}}=\frac{-3\sqrt{6}+9\sqrt{10}-7\sqrt{15}+105}{84}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}+7\sqrt{5}}{2\sqrt{3}+6\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}+7\sqrt{5}}{2\sqrt{3}+6\sqrt{5}}\frac{2\sqrt{3}-6\sqrt{5}}{2\sqrt{3}-6\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{6}-18\sqrt{10}+14\sqrt{15}-210}{12-12\sqrt{15}+12\sqrt{15}-180} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{6}-18\sqrt{10}+14\sqrt{15}-210}{-168} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{6}-9\sqrt{10}+7\sqrt{15}-105}{-84} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3\sqrt{6}+9\sqrt{10}-7\sqrt{15}+105}{84}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}- 6 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{2} + 7 \sqrt{5}\right) } \cdot \left( 2 \sqrt{3}- 6 \sqrt{5}\right) = \color{blue}{ 3 \sqrt{2}} \cdot 2 \sqrt{3}+\color{blue}{ 3 \sqrt{2}} \cdot- 6 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot 2 \sqrt{3}+\color{blue}{ 7 \sqrt{5}} \cdot- 6 \sqrt{5} = \\ = 6 \sqrt{6}- 18 \sqrt{10} + 14 \sqrt{15}-210 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 6 \sqrt{5}\right) } \cdot \left( 2 \sqrt{3}- 6 \sqrt{5}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot- 6 \sqrt{5}+\color{blue}{ 6 \sqrt{5}} \cdot 2 \sqrt{3}+\color{blue}{ 6 \sqrt{5}} \cdot- 6 \sqrt{5} = \\ = 12- 12 \sqrt{15} + 12 \sqrt{15}-180 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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