Question
Answer
$$\dfrac{2}{\sqrt{7}-5}=-\dfrac{\sqrt{7}+5}{9}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{\sqrt{7}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{\sqrt{7}-5}\frac{\sqrt{7}+5}{\sqrt{7}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{7}+10}{7+5\sqrt{7}-5\sqrt{7}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{7}+10}{-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{7}+5}{-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{7}+5}{9}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( \sqrt{7} + 5\right) = \color{blue}{2} \cdot \sqrt{7}+\color{blue}{2} \cdot5 = \\ = 2 \sqrt{7} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7}-5\right) } \cdot \left( \sqrt{7} + 5\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot5\color{blue}{-5} \cdot \sqrt{7}\color{blue}{-5} \cdot5 = \\ = 7 + 5 \sqrt{7}- 5 \sqrt{7}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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