Answer
$$\frac{6}{5-4\sqrt{10}}=-\frac{30+24\sqrt{10}}{135}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{5-4\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{5-4\sqrt{10}}\frac{5+4\sqrt{10}}{5+4\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30+24\sqrt{10}}{25+20\sqrt{10}-20\sqrt{10}-160} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30+24\sqrt{10}}{-135} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{30+24\sqrt{10}}{135}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 4 \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 5 + 4 \sqrt{10}\right) = \color{blue}{6} \cdot5+\color{blue}{6} \cdot 4 \sqrt{10} = \\ = 30 + 24 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 5- 4 \sqrt{10}\right) } \cdot \left( 5 + 4 \sqrt{10}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 4 \sqrt{10}\color{blue}{- 4 \sqrt{10}} \cdot5\color{blue}{- 4 \sqrt{10}} \cdot 4 \sqrt{10} = \\ = 25 + 20 \sqrt{10}- 20 \sqrt{10}-160 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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