Question
Answer
$$\dfrac{\sqrt{10}+\sqrt{7}}{\sqrt{10}-\sqrt{7}}=\dfrac{17+2\sqrt{70}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{10}+\sqrt{7}}{\sqrt{10}-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{10}+\sqrt{7}}{\sqrt{10}-\sqrt{7}}\frac{\sqrt{10}+\sqrt{7}}{\sqrt{10}+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+\sqrt{70}+\sqrt{70}+7}{10+\sqrt{70}-\sqrt{70}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{17+2\sqrt{70}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10} + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{10} + \sqrt{7}\right) } \cdot \left( \sqrt{10} + \sqrt{7}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot \sqrt{10}+\color{blue}{ \sqrt{7}} \cdot \sqrt{7} = \\ = 10 + \sqrt{70} + \sqrt{70} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10}- \sqrt{7}\right) } \cdot \left( \sqrt{10} + \sqrt{7}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot \sqrt{10}\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 10 + \sqrt{70}- \sqrt{70}-7 $$ |
| ③ | Simplify numerator and denominator |
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