Question
Answer
$$\dfrac{6}{4-\sqrt{10}}=4+\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{4-\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{4-\sqrt{10}}\frac{4+\sqrt{10}}{4+\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{24+6\sqrt{10}}{16+4\sqrt{10}-4\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{24+6\sqrt{10}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4+\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4+\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 4 + \sqrt{10}\right) = \color{blue}{6} \cdot4+\color{blue}{6} \cdot \sqrt{10} = \\ = 24 + 6 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{10}\right) } \cdot \left( 4 + \sqrt{10}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot4\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 16 + 4 \sqrt{10}- 4 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Remove 1 from denominator. |
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