Answer
$$\frac{4}{7+\sqrt{14}}=\frac{28-4\sqrt{14}}{35}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{7+\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{7+\sqrt{14}}\frac{7-\sqrt{14}}{7-\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28-4\sqrt{14}}{49-7\sqrt{14}+7\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{28-4\sqrt{14}}{35}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7- \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 7- \sqrt{14}\right) = \color{blue}{4} \cdot7+\color{blue}{4} \cdot- \sqrt{14} = \\ = 28- 4 \sqrt{14} $$ 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 |
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