Question
Answer
$$\dfrac{2\sqrt{5}}{-10+7\sqrt{2}}=-10\sqrt{5}-7\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}}{-10+7\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}}{-10+7\sqrt{2}}\frac{-10-7\sqrt{2}}{-10-7\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-20\sqrt{5}-14\sqrt{10}}{100+70\sqrt{2}-70\sqrt{2}-98} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-20\sqrt{5}-14\sqrt{10}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-10\sqrt{5}-7\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-10\sqrt{5}-7\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -10- 7 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 \sqrt{5} } \cdot \left( -10- 7 \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot-10+\color{blue}{ 2 \sqrt{5}} \cdot- 7 \sqrt{2} = \\ = - 20 \sqrt{5}- 14 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( -10 + 7 \sqrt{2}\right) } \cdot \left( -10- 7 \sqrt{2}\right) = \color{blue}{-10} \cdot-10\color{blue}{-10} \cdot- 7 \sqrt{2}+\color{blue}{ 7 \sqrt{2}} \cdot-10+\color{blue}{ 7 \sqrt{2}} \cdot- 7 \sqrt{2} = \\ = 100 + 70 \sqrt{2}- 70 \sqrt{2}-98 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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