Answer
$$\frac{\sqrt{15}+4}{\sqrt{14}+8}=\frac{-\sqrt{210}+8\sqrt{15}-4\sqrt{14}+32}{50}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{15}+4}{\sqrt{14}+8}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{15}+4}{\sqrt{14}+8}\frac{\sqrt{14}-8}{\sqrt{14}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{210}-8\sqrt{15}+4\sqrt{14}-32}{14-8\sqrt{14}+8\sqrt{14}-64} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{210}-8\sqrt{15}+4\sqrt{14}-32}{-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{210}+8\sqrt{15}-4\sqrt{14}+32}{50}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{14}-8} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{15} + 4\right) } \cdot \left( \sqrt{14}-8\right) = \color{blue}{ \sqrt{15}} \cdot \sqrt{14}+\color{blue}{ \sqrt{15}} \cdot-8+\color{blue}{4} \cdot \sqrt{14}+\color{blue}{4} \cdot-8 = \\ = \sqrt{210}- 8 \sqrt{15} + 4 \sqrt{14}-32 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{14} + 8\right) } \cdot \left( \sqrt{14}-8\right) = \color{blue}{ \sqrt{14}} \cdot \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot-8+\color{blue}{8} \cdot \sqrt{14}+\color{blue}{8} \cdot-8 = \\ = 14- 8 \sqrt{14} + 8 \sqrt{14}-64 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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