Answer
$$\frac{\sqrt{15}}{4\sqrt{5}-7}=\frac{20\sqrt{3}+7\sqrt{15}}{31}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{15}}{4\sqrt{5}-7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{15}}{4\sqrt{5}-7}\frac{4\sqrt{5}+7}{4\sqrt{5}+7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20\sqrt{3}+7\sqrt{15}}{80+28\sqrt{5}-28\sqrt{5}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20\sqrt{3}+7\sqrt{15}}{31}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{5} + 7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{15} } \cdot \left( 4 \sqrt{5} + 7\right) = \color{blue}{ \sqrt{15}} \cdot 4 \sqrt{5}+\color{blue}{ \sqrt{15}} \cdot7 = \\ = 20 \sqrt{3} + 7 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{5}-7\right) } \cdot \left( 4 \sqrt{5} + 7\right) = \color{blue}{ 4 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot7\color{blue}{-7} \cdot 4 \sqrt{5}\color{blue}{-7} \cdot7 = \\ = 80 + 28 \sqrt{5}- 28 \sqrt{5}-49 $$ |
| ③ | Simplify numerator and denominator |
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