Question
Answer
$$\dfrac{12+\sqrt{45}}{3\sqrt{5}-6}=6\sqrt{5}+13$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{12+\sqrt{45}}{3\sqrt{5}-6}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{12+\sqrt{45}}{3\sqrt{5}-6}\frac{3\sqrt{5}+6}{3\sqrt{5}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36\sqrt{5}+72+45+18\sqrt{5}}{45+18\sqrt{5}-18\sqrt{5}-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{54\sqrt{5}+117}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6\sqrt{5}+13}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}6\sqrt{5}+13\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 6} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 12 + \sqrt{45}\right) } \cdot \left( 3 \sqrt{5} + 6\right) = \color{blue}{12} \cdot 3 \sqrt{5}+\color{blue}{12} \cdot6+\color{blue}{ \sqrt{45}} \cdot 3 \sqrt{5}+\color{blue}{ \sqrt{45}} \cdot6 = \\ = 36 \sqrt{5} + 72 + 45 + 18 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}-6\right) } \cdot \left( 3 \sqrt{5} + 6\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot6\color{blue}{-6} \cdot 3 \sqrt{5}\color{blue}{-6} \cdot6 = \\ = 45 + 18 \sqrt{5}- 18 \sqrt{5}-36 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Remove 1 from denominator. |
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