Question
Answer
$$\dfrac{6\sqrt{5}}{9+5\sqrt{3}}=9\sqrt{5}-5\sqrt{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6\sqrt{5}}{9+5\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6\sqrt{5}}{9+5\sqrt{3}}\frac{9-5\sqrt{3}}{9-5\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{54\sqrt{5}-30\sqrt{15}}{81-45\sqrt{3}+45\sqrt{3}-75} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{54\sqrt{5}-30\sqrt{15}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9\sqrt{5}-5\sqrt{15}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}9\sqrt{5}-5\sqrt{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9- 5 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 \sqrt{5} } \cdot \left( 9- 5 \sqrt{3}\right) = \color{blue}{ 6 \sqrt{5}} \cdot9+\color{blue}{ 6 \sqrt{5}} \cdot- 5 \sqrt{3} = \\ = 54 \sqrt{5}- 30 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 9 + 5 \sqrt{3}\right) } \cdot \left( 9- 5 \sqrt{3}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot9+\color{blue}{ 5 \sqrt{3}} \cdot- 5 \sqrt{3} = \\ = 81- 45 \sqrt{3} + 45 \sqrt{3}-75 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Remove 1 from denominator. |
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