Answer
$$\frac{\sqrt{7}}{5\sqrt{5}-8}=\frac{5\sqrt{35}+8\sqrt{7}}{61}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{7}}{5\sqrt{5}-8}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}}{5\sqrt{5}-8}\frac{5\sqrt{5}+8}{5\sqrt{5}+8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{35}+8\sqrt{7}}{125+40\sqrt{5}-40\sqrt{5}-64} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{35}+8\sqrt{7}}{61}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{5} + 8} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{7} } \cdot \left( 5 \sqrt{5} + 8\right) = \color{blue}{ \sqrt{7}} \cdot 5 \sqrt{5}+\color{blue}{ \sqrt{7}} \cdot8 = \\ = 5 \sqrt{35} + 8 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{5}-8\right) } \cdot \left( 5 \sqrt{5} + 8\right) = \color{blue}{ 5 \sqrt{5}} \cdot 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot8\color{blue}{-8} \cdot 5 \sqrt{5}\color{blue}{-8} \cdot8 = \\ = 125 + 40 \sqrt{5}- 40 \sqrt{5}-64 $$ |
| ③ | Simplify numerator and denominator |
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