Answer
$$\frac{1+7\sqrt{2}}{5-3\sqrt{8}}=-\frac{89+41\sqrt{2}}{47}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+7\sqrt{2}}{5-3\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1+7\sqrt{2}}{5-3\sqrt{8}}\frac{5+3\sqrt{8}}{5+3\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+6\sqrt{2}+35\sqrt{2}+84}{25+30\sqrt{2}-30\sqrt{2}-72} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{89+41\sqrt{2}}{-47} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{89+41\sqrt{2}}{47}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 3 \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1 + 7 \sqrt{2}\right) } \cdot \left( 5 + 3 \sqrt{8}\right) = \color{blue}{1} \cdot5+\color{blue}{1} \cdot 3 \sqrt{8}+\color{blue}{ 7 \sqrt{2}} \cdot5+\color{blue}{ 7 \sqrt{2}} \cdot 3 \sqrt{8} = \\ = 5 + 6 \sqrt{2} + 35 \sqrt{2} + 84 $$ Simplify denominator. $$ \color{blue}{ \left( 5- 3 \sqrt{8}\right) } \cdot \left( 5 + 3 \sqrt{8}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 3 \sqrt{8}\color{blue}{- 3 \sqrt{8}} \cdot5\color{blue}{- 3 \sqrt{8}} \cdot 3 \sqrt{8} = \\ = 25 + 30 \sqrt{2}- 30 \sqrt{2}-72 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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