Question
Answer
$$\dfrac{3}{7-\sqrt{11}}=\dfrac{21+3\sqrt{11}}{38}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{7-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{7-\sqrt{11}}\frac{7+\sqrt{11}}{7+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{21+3\sqrt{11}}{49+7\sqrt{11}-7\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{21+3\sqrt{11}}{38}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 7 + \sqrt{11}\right) = \color{blue}{3} \cdot7+\color{blue}{3} \cdot \sqrt{11} = \\ = 21 + 3 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 7- \sqrt{11}\right) } \cdot \left( 7 + \sqrt{11}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot7\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 49 + 7 \sqrt{11}- 7 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
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