Answer
$$\frac{6+\sqrt{15}}{3-\sqrt{15}}=-\frac{11+3\sqrt{15}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6+\sqrt{15}}{3-\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6+\sqrt{15}}{3-\sqrt{15}}\frac{3+\sqrt{15}}{3+\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18+6\sqrt{15}+3\sqrt{15}+15}{9+3\sqrt{15}-3\sqrt{15}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{33+9\sqrt{15}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11+3\sqrt{15}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{11+3\sqrt{15}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 6 + \sqrt{15}\right) } \cdot \left( 3 + \sqrt{15}\right) = \color{blue}{6} \cdot3+\color{blue}{6} \cdot \sqrt{15}+\color{blue}{ \sqrt{15}} \cdot3+\color{blue}{ \sqrt{15}} \cdot \sqrt{15} = \\ = 18 + 6 \sqrt{15} + 3 \sqrt{15} + 15 $$ Simplify denominator. $$ \color{blue}{ \left( 3- \sqrt{15}\right) } \cdot \left( 3 + \sqrt{15}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot \sqrt{15}\color{blue}{- \sqrt{15}} \cdot3\color{blue}{- \sqrt{15}} \cdot \sqrt{15} = \\ = 9 + 3 \sqrt{15}- 3 \sqrt{15}-15 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator