Question
Answer
$$\dfrac{5}{3+\sqrt{45}}=\dfrac{-15+15\sqrt{5}}{36}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{3+\sqrt{45}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{3+\sqrt{45}}\frac{3-\sqrt{45}}{3-\sqrt{45}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15-15\sqrt{5}}{9-9\sqrt{5}+9\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15-15\sqrt{5}}{-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-15+15\sqrt{5}}{36}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{45}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 3- \sqrt{45}\right) = \color{blue}{5} \cdot3+\color{blue}{5} \cdot- \sqrt{45} = \\ = 15- 15 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{45}\right) } \cdot \left( 3- \sqrt{45}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{45}+\color{blue}{ \sqrt{45}} \cdot3+\color{blue}{ \sqrt{45}} \cdot- \sqrt{45} = \\ = 9- 9 \sqrt{5} + 9 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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