Answer
$$\frac{8}{8\sqrt{7}+\sqrt{5}}=\frac{64\sqrt{7}-8\sqrt{5}}{443}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{8\sqrt{7}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{8\sqrt{7}+\sqrt{5}}\frac{8\sqrt{7}-\sqrt{5}}{8\sqrt{7}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{64\sqrt{7}-8\sqrt{5}}{448-8\sqrt{35}+8\sqrt{35}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{64\sqrt{7}-8\sqrt{5}}{443}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8 \sqrt{7}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( 8 \sqrt{7}- \sqrt{5}\right) = \color{blue}{8} \cdot 8 \sqrt{7}+\color{blue}{8} \cdot- \sqrt{5} = \\ = 64 \sqrt{7}- 8 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 8 \sqrt{7} + \sqrt{5}\right) } \cdot \left( 8 \sqrt{7}- \sqrt{5}\right) = \color{blue}{ 8 \sqrt{7}} \cdot 8 \sqrt{7}+\color{blue}{ 8 \sqrt{7}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot 8 \sqrt{7}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 448- 8 \sqrt{35} + 8 \sqrt{35}-5 $$ |
| ③ | Simplify numerator and denominator |
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