Answer
$$\frac{5+3\sqrt{2}}{5-3\sqrt{2}}=\frac{43+30\sqrt{2}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5+3\sqrt{2}}{5-3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5+3\sqrt{2}}{5-3\sqrt{2}}\frac{5+3\sqrt{2}}{5+3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25+15\sqrt{2}+15\sqrt{2}+18}{25+15\sqrt{2}-15\sqrt{2}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{43+30\sqrt{2}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 + 3 \sqrt{2}\right) } \cdot \left( 5 + 3 \sqrt{2}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot5+\color{blue}{ 3 \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 25 + 15 \sqrt{2} + 15 \sqrt{2} + 18 $$ Simplify denominator. $$ \color{blue}{ \left( 5- 3 \sqrt{2}\right) } \cdot \left( 5 + 3 \sqrt{2}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 3 \sqrt{2}\color{blue}{- 3 \sqrt{2}} \cdot5\color{blue}{- 3 \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 25 + 15 \sqrt{2}- 15 \sqrt{2}-18 $$ |
| ③ | Simplify numerator and denominator |
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