Question
Answer
$$\dfrac{6+5\sqrt{2}}{4+3\sqrt{2}}=3-\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6+5\sqrt{2}}{4+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6+5\sqrt{2}}{4+3\sqrt{2}}\frac{4-3\sqrt{2}}{4-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{24-18\sqrt{2}+20\sqrt{2}-30}{16-12\sqrt{2}+12\sqrt{2}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-6+2\sqrt{2}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3+\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3-\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}3-\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 6 + 5 \sqrt{2}\right) } \cdot \left( 4- 3 \sqrt{2}\right) = \color{blue}{6} \cdot4+\color{blue}{6} \cdot- 3 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot4+\color{blue}{ 5 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 24- 18 \sqrt{2} + 20 \sqrt{2}-30 $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 3 \sqrt{2}\right) } \cdot \left( 4- 3 \sqrt{2}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot4+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 16- 12 \sqrt{2} + 12 \sqrt{2}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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