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