Question
Answer
$$\dfrac{5}{7-\sqrt{14}}=\dfrac{7+\sqrt{14}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{7-\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{7-\sqrt{14}}\frac{7+\sqrt{14}}{7+\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{35+5\sqrt{14}}{49+7\sqrt{14}-7\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{35+5\sqrt{14}}{35} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{7+\sqrt{14}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 7 + \sqrt{14}\right) = \color{blue}{5} \cdot7+\color{blue}{5} \cdot \sqrt{14} = \\ = 35 + 5 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 7- \sqrt{14}\right) } \cdot \left( 7 + \sqrt{14}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot \sqrt{14}\color{blue}{- \sqrt{14}} \cdot7\color{blue}{- \sqrt{14}} \cdot \sqrt{14} = \\ = 49 + 7 \sqrt{14}- 7 \sqrt{14}-14 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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