Question
Answer
$$\dfrac{2\sqrt{3}-3\sqrt{5}}{5\sqrt{6}+9\sqrt{8}}=\dfrac{-10\sqrt{2}+12\sqrt{6}+5\sqrt{30}-18\sqrt{10}}{166}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{3}-3\sqrt{5}}{5\sqrt{6}+9\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{3}-3\sqrt{5}}{5\sqrt{6}+9\sqrt{8}}\frac{5\sqrt{6}-9\sqrt{8}}{5\sqrt{6}-9\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30\sqrt{2}-36\sqrt{6}-15\sqrt{30}+54\sqrt{10}}{150-180\sqrt{3}+180\sqrt{3}-648} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30\sqrt{2}-36\sqrt{6}-15\sqrt{30}+54\sqrt{10}}{-498} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10\sqrt{2}-12\sqrt{6}-5\sqrt{30}+18\sqrt{10}}{-166} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-10\sqrt{2}+12\sqrt{6}+5\sqrt{30}-18\sqrt{10}}{166}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{6}- 9 \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{3}- 3 \sqrt{5}\right) } \cdot \left( 5 \sqrt{6}- 9 \sqrt{8}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 5 \sqrt{6}+\color{blue}{ 2 \sqrt{3}} \cdot- 9 \sqrt{8}\color{blue}{- 3 \sqrt{5}} \cdot 5 \sqrt{6}\color{blue}{- 3 \sqrt{5}} \cdot- 9 \sqrt{8} = \\ = 30 \sqrt{2}- 36 \sqrt{6}- 15 \sqrt{30} + 54 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{6} + 9 \sqrt{8}\right) } \cdot \left( 5 \sqrt{6}- 9 \sqrt{8}\right) = \color{blue}{ 5 \sqrt{6}} \cdot 5 \sqrt{6}+\color{blue}{ 5 \sqrt{6}} \cdot- 9 \sqrt{8}+\color{blue}{ 9 \sqrt{8}} \cdot 5 \sqrt{6}+\color{blue}{ 9 \sqrt{8}} \cdot- 9 \sqrt{8} = \\ = 150- 180 \sqrt{3} + 180 \sqrt{3}-648 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Multiply both numerator and denominator by -1. |
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