Answer
$$(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}})(\sqrt{6}+11)=-115$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}})(\sqrt{6}+11)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(\frac{15\sqrt{6}-15}{6-\sqrt{6}+\sqrt{6}-1}+\frac{4\sqrt{6}+8}{6+2\sqrt{6}-2\sqrt{6}-4}-\frac{36+12\sqrt{6}}{9+3\sqrt{6}-3\sqrt{6}-6})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(\frac{15\sqrt{6}-15}{5}+\frac{4\sqrt{6}+8}{2}-\frac{36+12\sqrt{6}}{3})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(\frac{30\sqrt{6}-30+20\sqrt{6}+40}{10}-\frac{36+12\sqrt{6}}{3})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(\frac{50\sqrt{6}+10}{10}-\frac{36+12\sqrt{6}}{3})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(\frac{5\sqrt{6}+1}{1}-\frac{36+12\sqrt{6}}{3})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(5\sqrt{6}+1-\frac{36+12\sqrt{6}}{3})(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{15\sqrt{6}+3-36-12\sqrt{6}}{3}(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{3\sqrt{6}-33}{3}(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{\sqrt{6}-11}{1}(\sqrt{6}+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}(\sqrt{6}-11)(\sqrt{6}+11) \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}6+11\sqrt{6}-11\sqrt{6}-121 \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}-115\end{aligned} $$ | |
| ① | Multiply in a numerator. $$ \color{blue}{ 15 } \cdot \left( \sqrt{6}-1\right) = \color{blue}{15} \cdot \sqrt{6}+\color{blue}{15} \cdot-1 = \\ = 15 \sqrt{6}-15 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6} + 1\right) } \cdot \left( \sqrt{6}-1\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot-1+\color{blue}{1} \cdot \sqrt{6}+\color{blue}{1} \cdot-1 = \\ = 6- \sqrt{6} + \sqrt{6}-1 $$Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{6} + 2\right) = \color{blue}{4} \cdot \sqrt{6}+\color{blue}{4} \cdot2 = \\ = 4 \sqrt{6} + 8 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6}-2\right) } \cdot \left( \sqrt{6} + 2\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot2\color{blue}{-2} \cdot \sqrt{6}\color{blue}{-2} \cdot2 = \\ = 6 + 2 \sqrt{6}- 2 \sqrt{6}-4 $$Multiply in a numerator. $$ \color{blue}{ 12 } \cdot \left( 3 + \sqrt{6}\right) = \color{blue}{12} \cdot3+\color{blue}{12} \cdot \sqrt{6} = \\ = 36 + 12 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 3- \sqrt{6}\right) } \cdot \left( 3 + \sqrt{6}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot \sqrt{6}\color{blue}{- \sqrt{6}} \cdot3\color{blue}{- \sqrt{6}} \cdot \sqrt{6} = \\ = 9 + 3 \sqrt{6}- 3 \sqrt{6}-6 $$ |
| ② | Simplify numerator and denominatorSimplify numerator and denominatorSimplify numerator and denominator |
| ③ | $$ \frac{15\sqrt{6}-15}{5}+\frac{4\sqrt{6}+8}{2}
= \frac{15\sqrt{6}-15}{5} \cdot \color{blue}{\frac{ 2 }{ 2}} + \frac{4\sqrt{6}+8}{2} \cdot \color{blue}{\frac{ 5 }{ 5}}
= \frac{30\sqrt{6}-30+20\sqrt{6}+40}{10} $$ |
| ④ | Simplify numerator and denominator |
| ⑤ | Divide both numerator and denominator by 10. |
| ⑥ | Remove 1 from denominator. |
| ⑦ | $$ 5\sqrt{6}+1-\frac{36+12\sqrt{6}}{3}
= 5\sqrt{6}+1 \cdot \color{blue}{\frac{ 3 }{ 3}} - \frac{36+12\sqrt{6}}{3} \cdot \color{blue}{\frac{ 1 }{ 1}}
= \frac{15\sqrt{6}+3-36-12\sqrt{6}}{3} $$ |
| ⑧ | Simplify numerator and denominator |
| ⑨ | Divide both numerator and denominator by 3. |
| ⑩ | Remove 1 from denominator. |
| ⑪ | $$ \color{blue}{ \left( \sqrt{6}-11\right) } \cdot \left( \sqrt{6} + 11\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot11\color{blue}{-11} \cdot \sqrt{6}\color{blue}{-11} \cdot11 = \\ = 6 + 11 \sqrt{6}- 11 \sqrt{6}-121 $$ |
| ⑫ | Combine like terms |
This page was created using
Simplify Radical Expression