Answer
$$\frac{14}{2\sqrt{2}-1}=\frac{28\sqrt{2}+14}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{14}{2\sqrt{2}-1}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{14}{2\sqrt{2}-1}\frac{2\sqrt{2}+1}{2\sqrt{2}+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28\sqrt{2}+14}{8+2\sqrt{2}-2\sqrt{2}-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{28\sqrt{2}+14}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{2} + 1} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 14 } \cdot \left( 2 \sqrt{2} + 1\right) = \color{blue}{14} \cdot 2 \sqrt{2}+\color{blue}{14} \cdot1 = \\ = 28 \sqrt{2} + 14 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{2}-1\right) } \cdot \left( 2 \sqrt{2} + 1\right) = \color{blue}{ 2 \sqrt{2}} \cdot 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot1\color{blue}{-1} \cdot 2 \sqrt{2}\color{blue}{-1} \cdot1 = \\ = 8 + 2 \sqrt{2}- 2 \sqrt{2}-1 $$ |
| ③ | Simplify numerator and denominator |
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