Answer
$$\frac{\sqrt{5}}{\sqrt{45}-\sqrt{15}}=\frac{3+\sqrt{3}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{\sqrt{45}-\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{\sqrt{45}-\sqrt{15}}\frac{\sqrt{45}+\sqrt{15}}{\sqrt{45}+\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15+5\sqrt{3}}{45+15\sqrt{3}-15\sqrt{3}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15+5\sqrt{3}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3+\sqrt{3}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{45} + \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( \sqrt{45} + \sqrt{15}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{45}+\color{blue}{ \sqrt{5}} \cdot \sqrt{15} = \\ = 15 + 5 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{45}- \sqrt{15}\right) } \cdot \left( \sqrt{45} + \sqrt{15}\right) = \color{blue}{ \sqrt{45}} \cdot \sqrt{45}+\color{blue}{ \sqrt{45}} \cdot \sqrt{15}\color{blue}{- \sqrt{15}} \cdot \sqrt{45}\color{blue}{- \sqrt{15}} \cdot \sqrt{15} = \\ = 45 + 15 \sqrt{3}- 15 \sqrt{3}-15 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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