Question
Answer
$$\dfrac{3\sqrt{3}+7}{3\sqrt{3}-5}=31+18\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{3}+7}{3\sqrt{3}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{3}+7}{3\sqrt{3}-5}\frac{3\sqrt{3}+5}{3\sqrt{3}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{27+15\sqrt{3}+21\sqrt{3}+35}{27+15\sqrt{3}-15\sqrt{3}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{62+36\sqrt{3}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{31+18\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}31+18\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{3} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{3} + 7\right) } \cdot \left( 3 \sqrt{3} + 5\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot5+\color{blue}{7} \cdot 3 \sqrt{3}+\color{blue}{7} \cdot5 = \\ = 27 + 15 \sqrt{3} + 21 \sqrt{3} + 35 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{3}-5\right) } \cdot \left( 3 \sqrt{3} + 5\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot5\color{blue}{-5} \cdot 3 \sqrt{3}\color{blue}{-5} \cdot5 = \\ = 27 + 15 \sqrt{3}- 15 \sqrt{3}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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