Question
Answer
$$\dfrac{1+7\sqrt{2}}{10-3\sqrt{8}}=\dfrac{47+38\sqrt{2}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+7\sqrt{2}}{10-3\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1+7\sqrt{2}}{10-3\sqrt{8}}\frac{10+3\sqrt{8}}{10+3\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+6\sqrt{2}+70\sqrt{2}+84}{100+60\sqrt{2}-60\sqrt{2}-72} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{94+76\sqrt{2}}{28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{47+38\sqrt{2}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10 + 3 \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1 + 7 \sqrt{2}\right) } \cdot \left( 10 + 3 \sqrt{8}\right) = \color{blue}{1} \cdot10+\color{blue}{1} \cdot 3 \sqrt{8}+\color{blue}{ 7 \sqrt{2}} \cdot10+\color{blue}{ 7 \sqrt{2}} \cdot 3 \sqrt{8} = \\ = 10 + 6 \sqrt{2} + 70 \sqrt{2} + 84 $$ Simplify denominator. $$ \color{blue}{ \left( 10- 3 \sqrt{8}\right) } \cdot \left( 10 + 3 \sqrt{8}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot 3 \sqrt{8}\color{blue}{- 3 \sqrt{8}} \cdot10\color{blue}{- 3 \sqrt{8}} \cdot 3 \sqrt{8} = \\ = 100 + 60 \sqrt{2}- 60 \sqrt{2}-72 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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