Question
Answer
$$\dfrac{3\sqrt{2}+\sqrt{5}}{3\sqrt{6}-6\sqrt{9}}=-\dfrac{6\sqrt{3}+18\sqrt{2}+\sqrt{30}+6\sqrt{5}}{90}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}+\sqrt{5}}{3\sqrt{6}-6\sqrt{9}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}+\sqrt{5}}{3\sqrt{6}-6\sqrt{9}}\frac{3\sqrt{6}+6\sqrt{9}}{3\sqrt{6}+6\sqrt{9}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18\sqrt{3}+54\sqrt{2}+3\sqrt{30}+18\sqrt{5}}{54+54\sqrt{6}-54\sqrt{6}-324} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18\sqrt{3}+54\sqrt{2}+3\sqrt{30}+18\sqrt{5}}{-270} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6\sqrt{3}+18\sqrt{2}+\sqrt{30}+6\sqrt{5}}{-90} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{6\sqrt{3}+18\sqrt{2}+\sqrt{30}+6\sqrt{5}}{90}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{6} + 6 \sqrt{9}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{2} + \sqrt{5}\right) } \cdot \left( 3 \sqrt{6} + 6 \sqrt{9}\right) = \color{blue}{ 3 \sqrt{2}} \cdot 3 \sqrt{6}+\color{blue}{ 3 \sqrt{2}} \cdot 6 \sqrt{9}+\color{blue}{ \sqrt{5}} \cdot 3 \sqrt{6}+\color{blue}{ \sqrt{5}} \cdot 6 \sqrt{9} = \\ = 18 \sqrt{3} + 54 \sqrt{2} + 3 \sqrt{30} + 18 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{6}- 6 \sqrt{9}\right) } \cdot \left( 3 \sqrt{6} + 6 \sqrt{9}\right) = \color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot 6 \sqrt{9}\color{blue}{- 6 \sqrt{9}} \cdot 3 \sqrt{6}\color{blue}{- 6 \sqrt{9}} \cdot 6 \sqrt{9} = \\ = 54 + 54 \sqrt{6}- 54 \sqrt{6}-324 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Place a negative sign in front of a fraction. |
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