Question
Answer
$$\dfrac{\sqrt{3}+\sqrt{7}}{2\sqrt{3}+4\sqrt{5}}=\dfrac{-3+2\sqrt{15}-\sqrt{21}+2\sqrt{35}}{34}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}+\sqrt{7}}{2\sqrt{3}+4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}+\sqrt{7}}{2\sqrt{3}+4\sqrt{5}}\frac{2\sqrt{3}-4\sqrt{5}}{2\sqrt{3}-4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6-4\sqrt{15}+2\sqrt{21}-4\sqrt{35}}{12-8\sqrt{15}+8\sqrt{15}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6-4\sqrt{15}+2\sqrt{21}-4\sqrt{35}}{-68} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-2\sqrt{15}+\sqrt{21}-2\sqrt{35}}{-34} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3+2\sqrt{15}-\sqrt{21}+2\sqrt{35}}{34}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}- 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{3} + \sqrt{7}\right) } \cdot \left( 2 \sqrt{3}- 4 \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot- 4 \sqrt{5}+\color{blue}{ \sqrt{7}} \cdot 2 \sqrt{3}+\color{blue}{ \sqrt{7}} \cdot- 4 \sqrt{5} = \\ = 6- 4 \sqrt{15} + 2 \sqrt{21}- 4 \sqrt{35} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 4 \sqrt{5}\right) } \cdot \left( 2 \sqrt{3}- 4 \sqrt{5}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot- 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot 2 \sqrt{3}+\color{blue}{ 4 \sqrt{5}} \cdot- 4 \sqrt{5} = \\ = 12- 8 \sqrt{15} + 8 \sqrt{15}-80 $$ |
| 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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