Answer
$$\frac{30}{5\sqrt{3}-3\sqrt{5}}=5\sqrt{3}+3\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{30}{5\sqrt{3}-3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{30}{5\sqrt{3}-3\sqrt{5}}\frac{5\sqrt{3}+3\sqrt{5}}{5\sqrt{3}+3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{150\sqrt{3}+90\sqrt{5}}{75+15\sqrt{15}-15\sqrt{15}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{150\sqrt{3}+90\sqrt{5}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{3}+3\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}5\sqrt{3}+3\sqrt{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3} + 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 30 } \cdot \left( 5 \sqrt{3} + 3 \sqrt{5}\right) = \color{blue}{30} \cdot 5 \sqrt{3}+\color{blue}{30} \cdot 3 \sqrt{5} = \\ = 150 \sqrt{3} + 90 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3}- 3 \sqrt{5}\right) } \cdot \left( 5 \sqrt{3} + 3 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot 3 \sqrt{5}\color{blue}{- 3 \sqrt{5}} \cdot 5 \sqrt{3}\color{blue}{- 3 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 75 + 15 \sqrt{15}- 15 \sqrt{15}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 30. |
| ⑤ | Remove 1 from denominator. |
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