Question
Answer
$$\dfrac{\sqrt{2}}{\sqrt{7}+3\sqrt{10}}=\dfrac{-\sqrt{14}+6\sqrt{5}}{83}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{\sqrt{7}+3\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{\sqrt{7}+3\sqrt{10}}\frac{\sqrt{7}-3\sqrt{10}}{\sqrt{7}-3\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{14}-6\sqrt{5}}{7-3\sqrt{70}+3\sqrt{70}-90} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{14}-6\sqrt{5}}{-83} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{14}+6\sqrt{5}}{83}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7}- 3 \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( \sqrt{7}- 3 \sqrt{10}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{7}+\color{blue}{ \sqrt{2}} \cdot- 3 \sqrt{10} = \\ = \sqrt{14}- 6 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7} + 3 \sqrt{10}\right) } \cdot \left( \sqrt{7}- 3 \sqrt{10}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot- 3 \sqrt{10}+\color{blue}{ 3 \sqrt{10}} \cdot \sqrt{7}+\color{blue}{ 3 \sqrt{10}} \cdot- 3 \sqrt{10} = \\ = 7- 3 \sqrt{70} + 3 \sqrt{70}-90 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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