Answer
$$\frac{\sqrt{30}+3\sqrt{3}}{\sqrt{30}-3\sqrt{3}}=19+6\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{30}+3\sqrt{3}}{\sqrt{30}-3\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{30}+3\sqrt{3}}{\sqrt{30}-3\sqrt{3}}\frac{\sqrt{30}+3\sqrt{3}}{\sqrt{30}+3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30+9\sqrt{10}+9\sqrt{10}+27}{30+9\sqrt{10}-9\sqrt{10}-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{57+18\sqrt{10}}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{19+6\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}19+6\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{30} + 3 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{30} + 3 \sqrt{3}\right) } \cdot \left( \sqrt{30} + 3 \sqrt{3}\right) = \color{blue}{ \sqrt{30}} \cdot \sqrt{30}+\color{blue}{ \sqrt{30}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot \sqrt{30}+\color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 30 + 9 \sqrt{10} + 9 \sqrt{10} + 27 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{30}- 3 \sqrt{3}\right) } \cdot \left( \sqrt{30} + 3 \sqrt{3}\right) = \color{blue}{ \sqrt{30}} \cdot \sqrt{30}+\color{blue}{ \sqrt{30}} \cdot 3 \sqrt{3}\color{blue}{- 3 \sqrt{3}} \cdot \sqrt{30}\color{blue}{- 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 30 + 9 \sqrt{10}- 9 \sqrt{10}-27 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Remove 1 from denominator. |
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