Question
Answer
$$\dfrac{2\sqrt{3}}{7-2\sqrt{10}}=\dfrac{14\sqrt{3}+4\sqrt{30}}{9}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{3}}{7-2\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{3}}{7-2\sqrt{10}}\frac{7+2\sqrt{10}}{7+2\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{14\sqrt{3}+4\sqrt{30}}{49+14\sqrt{10}-14\sqrt{10}-40} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14\sqrt{3}+4\sqrt{30}}{9}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + 2 \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 \sqrt{3} } \cdot \left( 7 + 2 \sqrt{10}\right) = \color{blue}{ 2 \sqrt{3}} \cdot7+\color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{10} = \\ = 14 \sqrt{3} + 4 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( 7- 2 \sqrt{10}\right) } \cdot \left( 7 + 2 \sqrt{10}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot 2 \sqrt{10}\color{blue}{- 2 \sqrt{10}} \cdot7\color{blue}{- 2 \sqrt{10}} \cdot 2 \sqrt{10} = \\ = 49 + 14 \sqrt{10}- 14 \sqrt{10}-40 $$ |
| ③ | Simplify numerator and denominator |
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