Question
Answer
$$\dfrac{30}{5\sqrt{3}-30\sqrt{5}}=-\dfrac{150\sqrt{3}+900\sqrt{5}}{4425}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{30}{5\sqrt{3}-30\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{30}{5\sqrt{3}-30\sqrt{5}}\frac{5\sqrt{3}+30\sqrt{5}}{5\sqrt{3}+30\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{150\sqrt{3}+900\sqrt{5}}{75+150\sqrt{15}-150\sqrt{15}-4500} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{150\sqrt{3}+900\sqrt{5}}{-4425} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{150\sqrt{3}+900\sqrt{5}}{4425}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3} + 30 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 30 } \cdot \left( 5 \sqrt{3} + 30 \sqrt{5}\right) = \color{blue}{30} \cdot 5 \sqrt{3}+\color{blue}{30} \cdot 30 \sqrt{5} = \\ = 150 \sqrt{3} + 900 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3}- 30 \sqrt{5}\right) } \cdot \left( 5 \sqrt{3} + 30 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot 30 \sqrt{5}\color{blue}{- 30 \sqrt{5}} \cdot 5 \sqrt{3}\color{blue}{- 30 \sqrt{5}} \cdot 30 \sqrt{5} = \\ = 75 + 150 \sqrt{15}- 150 \sqrt{15}-4500 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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