Question
Answer
$$\dfrac{3\sqrt{2}+4}{5\sqrt{2}-7}=58+41\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}+4}{5\sqrt{2}-7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}+4}{5\sqrt{2}-7}\frac{5\sqrt{2}+7}{5\sqrt{2}+7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30+21\sqrt{2}+20\sqrt{2}+28}{50+35\sqrt{2}-35\sqrt{2}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{58+41\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}58+41\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{2} + 7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{2} + 4\right) } \cdot \left( 5 \sqrt{2} + 7\right) = \color{blue}{ 3 \sqrt{2}} \cdot 5 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot7+\color{blue}{4} \cdot 5 \sqrt{2}+\color{blue}{4} \cdot7 = \\ = 30 + 21 \sqrt{2} + 20 \sqrt{2} + 28 $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{2}-7\right) } \cdot \left( 5 \sqrt{2} + 7\right) = \color{blue}{ 5 \sqrt{2}} \cdot 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot7\color{blue}{-7} \cdot 5 \sqrt{2}\color{blue}{-7} \cdot7 = \\ = 50 + 35 \sqrt{2}- 35 \sqrt{2}-49 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Remove 1 from denominator. |
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