Answer
$$\frac{6}{3+\sqrt{30}}=\frac{-18+6\sqrt{30}}{21}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{3+\sqrt{30}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{3+\sqrt{30}}\frac{3-\sqrt{30}}{3-\sqrt{30}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18-6\sqrt{30}}{9-3\sqrt{30}+3\sqrt{30}-30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18-6\sqrt{30}}{-21} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-18+6\sqrt{30}}{21}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{30}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 3- \sqrt{30}\right) = \color{blue}{6} \cdot3+\color{blue}{6} \cdot- \sqrt{30} = \\ = 18- 6 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{30}\right) } \cdot \left( 3- \sqrt{30}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{30}+\color{blue}{ \sqrt{30}} \cdot3+\color{blue}{ \sqrt{30}} \cdot- \sqrt{30} = \\ = 9- 3 \sqrt{30} + 3 \sqrt{30}-30 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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