Question
Answer
$$\dfrac{5+\sqrt{10}}{\sqrt{10}-3}=8\sqrt{10}+25$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5+\sqrt{10}}{\sqrt{10}-3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5+\sqrt{10}}{\sqrt{10}-3}\frac{\sqrt{10}+3}{\sqrt{10}+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{10}+15+10+3\sqrt{10}}{10+3\sqrt{10}-3\sqrt{10}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{10}+25}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8\sqrt{10}+25\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10} + 3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 + \sqrt{10}\right) } \cdot \left( \sqrt{10} + 3\right) = \color{blue}{5} \cdot \sqrt{10}+\color{blue}{5} \cdot3+\color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot3 = \\ = 5 \sqrt{10} + 15 + 10 + 3 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10}-3\right) } \cdot \left( \sqrt{10} + 3\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot3\color{blue}{-3} \cdot \sqrt{10}\color{blue}{-3} \cdot3 = \\ = 10 + 3 \sqrt{10}- 3 \sqrt{10}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Remove 1 from denominator. |
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