Question
Answer
$$\dfrac{5\sqrt{2}+3\sqrt{3}}{5+5\sqrt{5}}=\dfrac{-5\sqrt{2}+5\sqrt{10}-3\sqrt{3}+3\sqrt{15}}{20}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}+3\sqrt{3}}{5+5\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}+3\sqrt{3}}{5+5\sqrt{5}}\frac{5-5\sqrt{5}}{5-5\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25\sqrt{2}-25\sqrt{10}+15\sqrt{3}-15\sqrt{15}}{25-25\sqrt{5}+25\sqrt{5}-125} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25\sqrt{2}-25\sqrt{10}+15\sqrt{3}-15\sqrt{15}}{-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{2}-5\sqrt{10}+3\sqrt{3}-3\sqrt{15}}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-5\sqrt{2}+5\sqrt{10}-3\sqrt{3}+3\sqrt{15}}{20}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 5 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{2} + 3 \sqrt{3}\right) } \cdot \left( 5- 5 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{2}} \cdot5+\color{blue}{ 5 \sqrt{2}} \cdot- 5 \sqrt{5}+\color{blue}{ 3 \sqrt{3}} \cdot5+\color{blue}{ 3 \sqrt{3}} \cdot- 5 \sqrt{5} = \\ = 25 \sqrt{2}- 25 \sqrt{10} + 15 \sqrt{3}- 15 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 5 \sqrt{5}\right) } \cdot \left( 5- 5 \sqrt{5}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot5+\color{blue}{ 5 \sqrt{5}} \cdot- 5 \sqrt{5} = \\ = 25- 25 \sqrt{5} + 25 \sqrt{5}-125 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Multiply both numerator and denominator by -1. |
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