Question
Answer
$$\dfrac{5}{3\sqrt{5}+\sqrt{2}}=\dfrac{15\sqrt{5}-5\sqrt{2}}{43}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{3\sqrt{5}+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{3\sqrt{5}+\sqrt{2}}\frac{3\sqrt{5}-\sqrt{2}}{3\sqrt{5}-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15\sqrt{5}-5\sqrt{2}}{45-3\sqrt{10}+3\sqrt{10}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{5}-5\sqrt{2}}{43}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5}- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 3 \sqrt{5}- \sqrt{2}\right) = \color{blue}{5} \cdot 3 \sqrt{5}+\color{blue}{5} \cdot- \sqrt{2} = \\ = 15 \sqrt{5}- 5 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5} + \sqrt{2}\right) } \cdot \left( 3 \sqrt{5}- \sqrt{2}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 3 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 45- 3 \sqrt{10} + 3 \sqrt{10}-2 $$ |
| ③ | Simplify numerator and denominator |
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