Question
Answer
$$\dfrac{4+3\sqrt{2}}{5-3\sqrt{5}}=-\dfrac{20+12\sqrt{5}+15\sqrt{2}+9\sqrt{10}}{20}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+3\sqrt{2}}{5-3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+3\sqrt{2}}{5-3\sqrt{5}}\frac{5+3\sqrt{5}}{5+3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20+12\sqrt{5}+15\sqrt{2}+9\sqrt{10}}{25+15\sqrt{5}-15\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20+12\sqrt{5}+15\sqrt{2}+9\sqrt{10}}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{20+12\sqrt{5}+15\sqrt{2}+9\sqrt{10}}{20}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + 3 \sqrt{2}\right) } \cdot \left( 5 + 3 \sqrt{5}\right) = \color{blue}{4} \cdot5+\color{blue}{4} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot5+\color{blue}{ 3 \sqrt{2}} \cdot 3 \sqrt{5} = \\ = 20 + 12 \sqrt{5} + 15 \sqrt{2} + 9 \sqrt{10} $$ 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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