Answer
$$\frac{7\sqrt{5}}{5-3\sqrt{5}}=-\frac{35\sqrt{5}+105}{20}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{5}}{5-3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{5}}{5-3\sqrt{5}}\frac{5+3\sqrt{5}}{5+3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{35\sqrt{5}+105}{25+15\sqrt{5}-15\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{35\sqrt{5}+105}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{35\sqrt{5}+105}{20}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 \sqrt{5} } \cdot \left( 5 + 3 \sqrt{5}\right) = \color{blue}{ 7 \sqrt{5}} \cdot5+\color{blue}{ 7 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 35 \sqrt{5} + 105 $$ Simplify denominator. $$ \color{blue}{ \left( 5- 3 \sqrt{5}\right) } \cdot \left( 5 + 3 \sqrt{5}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 3 \sqrt{5}\color{blue}{- 3 \sqrt{5}} \cdot5\color{blue}{- 3 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 25 + 15 \sqrt{5}- 15 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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