Question
Answer
$$\dfrac{3+\sqrt{2}}{\sqrt{7}-9}=-\dfrac{3\sqrt{7}+27+\sqrt{14}+9\sqrt{2}}{74}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3+\sqrt{2}}{\sqrt{7}-9}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+\sqrt{2}}{\sqrt{7}-9}\frac{\sqrt{7}+9}{\sqrt{7}+9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{7}+27+\sqrt{14}+9\sqrt{2}}{7+9\sqrt{7}-9\sqrt{7}-81} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3\sqrt{7}+27+\sqrt{14}+9\sqrt{2}}{-74} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{3\sqrt{7}+27+\sqrt{14}+9\sqrt{2}}{74}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7} + 9} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 + \sqrt{2}\right) } \cdot \left( \sqrt{7} + 9\right) = \color{blue}{3} \cdot \sqrt{7}+\color{blue}{3} \cdot9+\color{blue}{ \sqrt{2}} \cdot \sqrt{7}+\color{blue}{ \sqrt{2}} \cdot9 = \\ = 3 \sqrt{7} + 27 + \sqrt{14} + 9 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7}-9\right) } \cdot \left( \sqrt{7} + 9\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot9\color{blue}{-9} \cdot \sqrt{7}\color{blue}{-9} \cdot9 = \\ = 7 + 9 \sqrt{7}- 9 \sqrt{7}-81 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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