Question
Answer
$$\dfrac{15}{4+\sqrt{11}}-3\sqrt{11}=\dfrac{60-30\sqrt{11}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{15}{4+\sqrt{11}}-3\sqrt{11}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{60-15\sqrt{11}}{16-4\sqrt{11}+4\sqrt{11}-11}-3\sqrt{11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{60-15\sqrt{11}}{5}-3\sqrt{11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60-15\sqrt{11}-15\sqrt{11}}{5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{60-30\sqrt{11}}{5}\end{aligned} $$ | |
| ① | Multiply in a numerator. $$ \color{blue}{ 15 } \cdot \left( 4- \sqrt{11}\right) = \color{blue}{15} \cdot4+\color{blue}{15} \cdot- \sqrt{11} = \\ = 60- 15 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 4 + \sqrt{11}\right) } \cdot \left( 4- \sqrt{11}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot4+\color{blue}{ \sqrt{11}} \cdot- \sqrt{11} = \\ = 16- 4 \sqrt{11} + 4 \sqrt{11}-11 $$ |
| ② | Simplify numerator and denominator |
| ③ | $$ \frac{60-15\sqrt{11}}{5}-3\sqrt{11}
= \frac{60-15\sqrt{11}}{5} \cdot \color{blue}{\frac{ 1 }{ 1}} - 3\sqrt{11} \cdot \color{blue}{\frac{ 5 }{ 5}}
= \frac{60-15\sqrt{11}-15\sqrt{11}}{5} $$ |
| ④ | Simplify numerator and denominator |
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