Question
Answer
$$\dfrac{5\sqrt{3}-7\sqrt{5}}{2\sqrt{5}+4\sqrt{3}}=\dfrac{-19\sqrt{15}+65}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{3}-7\sqrt{5}}{2\sqrt{5}+4\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{3}-7\sqrt{5}}{2\sqrt{5}+4\sqrt{3}}\frac{2\sqrt{5}-4\sqrt{3}}{2\sqrt{5}-4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10\sqrt{15}-60-70+28\sqrt{15}}{20-8\sqrt{15}+8\sqrt{15}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{38\sqrt{15}-130}{-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{19\sqrt{15}-65}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-19\sqrt{15}+65}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- 4 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{3}- 7 \sqrt{5}\right) } \cdot \left( 2 \sqrt{5}- 4 \sqrt{3}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 2 \sqrt{5}+\color{blue}{ 5 \sqrt{3}} \cdot- 4 \sqrt{3}\color{blue}{- 7 \sqrt{5}} \cdot 2 \sqrt{5}\color{blue}{- 7 \sqrt{5}} \cdot- 4 \sqrt{3} = \\ = 10 \sqrt{15}-60-70 + 28 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + 4 \sqrt{3}\right) } \cdot \left( 2 \sqrt{5}- 4 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot 2 \sqrt{5}+\color{blue}{ 4 \sqrt{3}} \cdot- 4 \sqrt{3} = \\ = 20- 8 \sqrt{15} + 8 \sqrt{15}-48 $$ |
| 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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