Question
Answer
$$\dfrac{\sqrt{14}}{13+\sqrt{15}}=\dfrac{13\sqrt{14}-\sqrt{210}}{154}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{14}}{13+\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{14}}{13+\sqrt{15}}\frac{13-\sqrt{15}}{13-\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{13\sqrt{14}-\sqrt{210}}{169-13\sqrt{15}+13\sqrt{15}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{13\sqrt{14}-\sqrt{210}}{154}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 13- \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{14} } \cdot \left( 13- \sqrt{15}\right) = \color{blue}{ \sqrt{14}} \cdot13+\color{blue}{ \sqrt{14}} \cdot- \sqrt{15} = \\ = 13 \sqrt{14}- \sqrt{210} $$ Simplify denominator. $$ \color{blue}{ \left( 13 + \sqrt{15}\right) } \cdot \left( 13- \sqrt{15}\right) = \color{blue}{13} \cdot13+\color{blue}{13} \cdot- \sqrt{15}+\color{blue}{ \sqrt{15}} \cdot13+\color{blue}{ \sqrt{15}} \cdot- \sqrt{15} = \\ = 169- 13 \sqrt{15} + 13 \sqrt{15}-15 $$ |
| ③ | Simplify numerator and denominator |
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