Question
Answer
$$\dfrac{3+\sqrt{7}}{3-4\sqrt{7}}=-\dfrac{37+15\sqrt{7}}{103}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3+\sqrt{7}}{3-4\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+\sqrt{7}}{3-4\sqrt{7}}\frac{3+4\sqrt{7}}{3+4\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9+12\sqrt{7}+3\sqrt{7}+28}{9+12\sqrt{7}-12\sqrt{7}-112} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{37+15\sqrt{7}}{-103} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{37+15\sqrt{7}}{103}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + 4 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 + \sqrt{7}\right) } \cdot \left( 3 + 4 \sqrt{7}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 4 \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot3+\color{blue}{ \sqrt{7}} \cdot 4 \sqrt{7} = \\ = 9 + 12 \sqrt{7} + 3 \sqrt{7} + 28 $$ Simplify denominator. $$ \color{blue}{ \left( 3- 4 \sqrt{7}\right) } \cdot \left( 3 + 4 \sqrt{7}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 4 \sqrt{7}\color{blue}{- 4 \sqrt{7}} \cdot3\color{blue}{- 4 \sqrt{7}} \cdot 4 \sqrt{7} = \\ = 9 + 12 \sqrt{7}- 12 \sqrt{7}-112 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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