Answer
$$\frac{1}{7+\sqrt{6}}=\frac{7-\sqrt{6}}{43}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{7+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{7+\sqrt{6}}\frac{7-\sqrt{6}}{7-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7-\sqrt{6}}{49-7\sqrt{6}+7\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{7-\sqrt{6}}{43}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 7- \sqrt{6}\right) = \color{blue}{1} \cdot7+\color{blue}{1} \cdot- \sqrt{6} = \\ = 7- \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 7 + \sqrt{6}\right) } \cdot \left( 7- \sqrt{6}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot7+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 49- 7 \sqrt{6} + 7 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
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