Answer
$$\frac{3+\sqrt{7}}{2-\sqrt{10}}=-\frac{6+3\sqrt{10}+2\sqrt{7}+\sqrt{70}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3+\sqrt{7}}{2-\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+\sqrt{7}}{2-\sqrt{10}}\frac{2+\sqrt{10}}{2+\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+3\sqrt{10}+2\sqrt{7}+\sqrt{70}}{4+2\sqrt{10}-2\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6+3\sqrt{10}+2\sqrt{7}+\sqrt{70}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{6+3\sqrt{10}+2\sqrt{7}+\sqrt{70}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 + \sqrt{7}\right) } \cdot \left( 2 + \sqrt{10}\right) = \color{blue}{3} \cdot2+\color{blue}{3} \cdot \sqrt{10}+\color{blue}{ \sqrt{7}} \cdot2+\color{blue}{ \sqrt{7}} \cdot \sqrt{10} = \\ = 6 + 3 \sqrt{10} + 2 \sqrt{7} + \sqrt{70} $$ Simplify denominator. $$ \color{blue}{ \left( 2- \sqrt{10}\right) } \cdot \left( 2 + \sqrt{10}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot2\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 4 + 2 \sqrt{10}- 2 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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