Answer
$$\frac{8}{-7+\sqrt{3}}=\frac{-56-8\sqrt{3}}{46}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{-7+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{-7+\sqrt{3}}\frac{-7-\sqrt{3}}{-7-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-56-8\sqrt{3}}{49+7\sqrt{3}-7\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-56-8\sqrt{3}}{46}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -7- \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( -7- \sqrt{3}\right) = \color{blue}{8} \cdot-7+\color{blue}{8} \cdot- \sqrt{3} = \\ = -56- 8 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( -7 + \sqrt{3}\right) } \cdot \left( -7- \sqrt{3}\right) = \color{blue}{-7} \cdot-7\color{blue}{-7} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot-7+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 49 + 7 \sqrt{3}- 7 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
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