Question
Answer
$$\dfrac{\sqrt{3}}{10+7\sqrt{7}}=\dfrac{-10\sqrt{3}+7\sqrt{21}}{243}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}}{10+7\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}}{10+7\sqrt{7}}\frac{10-7\sqrt{7}}{10-7\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10\sqrt{3}-7\sqrt{21}}{100-70\sqrt{7}+70\sqrt{7}-343} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{10\sqrt{3}-7\sqrt{21}}{-243} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-10\sqrt{3}+7\sqrt{21}}{243}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10- 7 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{3} } \cdot \left( 10- 7 \sqrt{7}\right) = \color{blue}{ \sqrt{3}} \cdot10+\color{blue}{ \sqrt{3}} \cdot- 7 \sqrt{7} = \\ = 10 \sqrt{3}- 7 \sqrt{21} $$ Simplify denominator. $$ \color{blue}{ \left( 10 + 7 \sqrt{7}\right) } \cdot \left( 10- 7 \sqrt{7}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot- 7 \sqrt{7}+\color{blue}{ 7 \sqrt{7}} \cdot10+\color{blue}{ 7 \sqrt{7}} \cdot- 7 \sqrt{7} = \\ = 100- 70 \sqrt{7} + 70 \sqrt{7}-343 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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