Question
Answer
$$\dfrac{\sqrt{45}+\sqrt{5}}{3+\sqrt{10}}=-12\sqrt{5}+20\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{45}+\sqrt{5}}{3+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{45}+\sqrt{5}}{3+\sqrt{10}}\frac{3-\sqrt{10}}{3-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{5}-15\sqrt{2}+3\sqrt{5}-5\sqrt{2}}{9-3\sqrt{10}+3\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{5}-20\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-12\sqrt{5}+20\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-12\sqrt{5}+20\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{45} + \sqrt{5}\right) } \cdot \left( 3- \sqrt{10}\right) = \color{blue}{ \sqrt{45}} \cdot3+\color{blue}{ \sqrt{45}} \cdot- \sqrt{10}+\color{blue}{ \sqrt{5}} \cdot3+\color{blue}{ \sqrt{5}} \cdot- \sqrt{10} = \\ = 9 \sqrt{5}- 15 \sqrt{2} + 3 \sqrt{5}- 5 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{10}\right) } \cdot \left( 3- \sqrt{10}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot3+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 9- 3 \sqrt{10} + 3 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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