Question
Answer
$$\dfrac{\sqrt{14}-\sqrt{7}}{14+\sqrt{7}}=\dfrac{2\sqrt{14}-\sqrt{2}-2\sqrt{7}+1}{27}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{14}-\sqrt{7}}{14+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{14}-\sqrt{7}}{14+\sqrt{7}}\frac{14-\sqrt{7}}{14-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{14\sqrt{14}-7\sqrt{2}-14\sqrt{7}+7}{196-14\sqrt{7}+14\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14\sqrt{14}-7\sqrt{2}-14\sqrt{7}+7}{189} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{14}-\sqrt{2}-2\sqrt{7}+1}{27}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 14- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{14}- \sqrt{7}\right) } \cdot \left( 14- \sqrt{7}\right) = \color{blue}{ \sqrt{14}} \cdot14+\color{blue}{ \sqrt{14}} \cdot- \sqrt{7}\color{blue}{- \sqrt{7}} \cdot14\color{blue}{- \sqrt{7}} \cdot- \sqrt{7} = \\ = 14 \sqrt{14}- 7 \sqrt{2}- 14 \sqrt{7} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( 14 + \sqrt{7}\right) } \cdot \left( 14- \sqrt{7}\right) = \color{blue}{14} \cdot14+\color{blue}{14} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot14+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 196- 14 \sqrt{7} + 14 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 7. |
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