Question
Answer
$$\dfrac{15}{\sqrt{7}-\sqrt{15}}=-\dfrac{15\sqrt{7}+15\sqrt{15}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{15}{\sqrt{7}-\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{15}{\sqrt{7}-\sqrt{15}}\frac{\sqrt{7}+\sqrt{15}}{\sqrt{7}+\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15\sqrt{7}+15\sqrt{15}}{7+\sqrt{105}-\sqrt{105}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{7}+15\sqrt{15}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{15\sqrt{7}+15\sqrt{15}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7} + \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 15 } \cdot \left( \sqrt{7} + \sqrt{15}\right) = \color{blue}{15} \cdot \sqrt{7}+\color{blue}{15} \cdot \sqrt{15} = \\ = 15 \sqrt{7} + 15 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7}- \sqrt{15}\right) } \cdot \left( \sqrt{7} + \sqrt{15}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot \sqrt{15}\color{blue}{- \sqrt{15}} \cdot \sqrt{7}\color{blue}{- \sqrt{15}} \cdot \sqrt{15} = \\ = 7 + \sqrt{105}- \sqrt{105}-15 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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