Answer
$$\frac{7}{\sqrt{8}-\sqrt{2}}=\frac{7\sqrt{2}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{\sqrt{8}-\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{\sqrt{8}-\sqrt{2}}\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{14\sqrt{2}+7\sqrt{2}}{8+4-4-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{21\sqrt{2}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{ 21 \sqrt{ 2 } : \color{blue}{ 3 } } { 6 : \color{blue}{ 3 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{7\sqrt{2}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{8} + \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( \sqrt{8} + \sqrt{2}\right) = \color{blue}{7} \cdot \sqrt{8}+\color{blue}{7} \cdot \sqrt{2} = \\ = 14 \sqrt{2} + 7 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{8}- \sqrt{2}\right) } \cdot \left( \sqrt{8} + \sqrt{2}\right) = \color{blue}{ \sqrt{8}} \cdot \sqrt{8}+\color{blue}{ \sqrt{8}} \cdot \sqrt{2}\color{blue}{- \sqrt{2}} \cdot \sqrt{8}\color{blue}{- \sqrt{2}} \cdot \sqrt{2} = \\ = 8 + 4-4-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide numerator and denominator by $ \color{blue}{ 3 } $. |
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