Question
Answer
$$\dfrac{5\sqrt{2}+2\sqrt{5}}{3\sqrt{6}-4\sqrt{5}}=-\dfrac{15\sqrt{3}+10\sqrt{10}+3\sqrt{30}+20}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}+2\sqrt{5}}{3\sqrt{6}-4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}+2\sqrt{5}}{3\sqrt{6}-4\sqrt{5}}\frac{3\sqrt{6}+4\sqrt{5}}{3\sqrt{6}+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30\sqrt{3}+20\sqrt{10}+6\sqrt{30}+40}{54+12\sqrt{30}-12\sqrt{30}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30\sqrt{3}+20\sqrt{10}+6\sqrt{30}+40}{-26} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{15\sqrt{3}+10\sqrt{10}+3\sqrt{30}+20}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{15\sqrt{3}+10\sqrt{10}+3\sqrt{30}+20}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{6} + 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{2} + 2 \sqrt{5}\right) } \cdot \left( 3 \sqrt{6} + 4 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{2}} \cdot 3 \sqrt{6}+\color{blue}{ 5 \sqrt{2}} \cdot 4 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot 3 \sqrt{6}+\color{blue}{ 2 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 30 \sqrt{3} + 20 \sqrt{10} + 6 \sqrt{30} + 40 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{6}- 4 \sqrt{5}\right) } \cdot \left( 3 \sqrt{6} + 4 \sqrt{5}\right) = \color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot 3 \sqrt{6}\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 54 + 12 \sqrt{30}- 12 \sqrt{30}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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