Answer
$$\frac{6-4\sqrt{2}}{6+4\sqrt{2}}=17-12\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6-4\sqrt{2}}{6+4\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6-4\sqrt{2}}{6+4\sqrt{2}}\frac{6-4\sqrt{2}}{6-4\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36-24\sqrt{2}-24\sqrt{2}+32}{36-24\sqrt{2}+24\sqrt{2}-32} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{68-48\sqrt{2}}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{17-12\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}17-12\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 4 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 6- 4 \sqrt{2}\right) } \cdot \left( 6- 4 \sqrt{2}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 4 \sqrt{2}\color{blue}{- 4 \sqrt{2}} \cdot6\color{blue}{- 4 \sqrt{2}} \cdot- 4 \sqrt{2} = \\ = 36- 24 \sqrt{2}- 24 \sqrt{2} + 32 $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 4 \sqrt{2}\right) } \cdot \left( 6- 4 \sqrt{2}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 4 \sqrt{2}+\color{blue}{ 4 \sqrt{2}} \cdot6+\color{blue}{ 4 \sqrt{2}} \cdot- 4 \sqrt{2} = \\ = 36- 24 \sqrt{2} + 24 \sqrt{2}-32 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Remove 1 from denominator. |
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