Question
Answer
$$\dfrac{3-2\sqrt{5}}{3\sqrt{3}+3}=\dfrac{3\sqrt{3}-3-2\sqrt{15}+2\sqrt{5}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-2\sqrt{5}}{3\sqrt{3}+3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3-2\sqrt{5}}{3\sqrt{3}+3}\frac{3\sqrt{3}-3}{3\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{3}-9-6\sqrt{15}+6\sqrt{5}}{27-9\sqrt{3}+9\sqrt{3}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{3}-9-6\sqrt{15}+6\sqrt{5}}{18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{3}-3-2\sqrt{15}+2\sqrt{5}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{3}-3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3- 2 \sqrt{5}\right) } \cdot \left( 3 \sqrt{3}-3\right) = \color{blue}{3} \cdot 3 \sqrt{3}+\color{blue}{3} \cdot-3\color{blue}{- 2 \sqrt{5}} \cdot 3 \sqrt{3}\color{blue}{- 2 \sqrt{5}} \cdot-3 = \\ = 9 \sqrt{3}-9- 6 \sqrt{15} + 6 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{3} + 3\right) } \cdot \left( 3 \sqrt{3}-3\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot-3+\color{blue}{3} \cdot 3 \sqrt{3}+\color{blue}{3} \cdot-3 = \\ = 27- 9 \sqrt{3} + 9 \sqrt{3}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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