Question
Answer
$$\dfrac{9\sqrt{10}}{\sqrt{2}-4\sqrt{7}}=-\dfrac{18\sqrt{5}+36\sqrt{70}}{110}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9\sqrt{10}}{\sqrt{2}-4\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9\sqrt{10}}{\sqrt{2}-4\sqrt{7}}\frac{\sqrt{2}+4\sqrt{7}}{\sqrt{2}+4\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18\sqrt{5}+36\sqrt{70}}{2+4\sqrt{14}-4\sqrt{14}-112} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18\sqrt{5}+36\sqrt{70}}{-110} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{18\sqrt{5}+36\sqrt{70}}{110}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2} + 4 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 \sqrt{10} } \cdot \left( \sqrt{2} + 4 \sqrt{7}\right) = \color{blue}{ 9 \sqrt{10}} \cdot \sqrt{2}+\color{blue}{ 9 \sqrt{10}} \cdot 4 \sqrt{7} = \\ = 18 \sqrt{5} + 36 \sqrt{70} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2}- 4 \sqrt{7}\right) } \cdot \left( \sqrt{2} + 4 \sqrt{7}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 4 \sqrt{7}\color{blue}{- 4 \sqrt{7}} \cdot \sqrt{2}\color{blue}{- 4 \sqrt{7}} \cdot 4 \sqrt{7} = \\ = 2 + 4 \sqrt{14}- 4 \sqrt{14}-112 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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