Answer
$$\frac{7}{5+\sqrt{29}}=\frac{-35+7\sqrt{29}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{5+\sqrt{29}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{5+\sqrt{29}}\frac{5-\sqrt{29}}{5-\sqrt{29}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{35-7\sqrt{29}}{25-5\sqrt{29}+5\sqrt{29}-29} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{35-7\sqrt{29}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-35+7\sqrt{29}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- \sqrt{29}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( 5- \sqrt{29}\right) = \color{blue}{7} \cdot5+\color{blue}{7} \cdot- \sqrt{29} = \\ = 35- 7 \sqrt{29} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + \sqrt{29}\right) } \cdot \left( 5- \sqrt{29}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{29}+\color{blue}{ \sqrt{29}} \cdot5+\color{blue}{ \sqrt{29}} \cdot- \sqrt{29} = \\ = 25- 5 \sqrt{29} + 5 \sqrt{29}-29 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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