Question
Answer
$$\dfrac{5\sqrt{2}+\sqrt{3}}{5+5\sqrt{2}}=\dfrac{-5\sqrt{2}+10-\sqrt{3}+\sqrt{6}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}+\sqrt{3}}{5+5\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}+\sqrt{3}}{5+5\sqrt{2}}\frac{5-5\sqrt{2}}{5-5\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25\sqrt{2}-50+5\sqrt{3}-5\sqrt{6}}{25-25\sqrt{2}+25\sqrt{2}-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25\sqrt{2}-50+5\sqrt{3}-5\sqrt{6}}{-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{2}-10+\sqrt{3}-\sqrt{6}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-5\sqrt{2}+10-\sqrt{3}+\sqrt{6}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 5 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{2} + \sqrt{3}\right) } \cdot \left( 5- 5 \sqrt{2}\right) = \color{blue}{ 5 \sqrt{2}} \cdot5+\color{blue}{ 5 \sqrt{2}} \cdot- 5 \sqrt{2}+\color{blue}{ \sqrt{3}} \cdot5+\color{blue}{ \sqrt{3}} \cdot- 5 \sqrt{2} = \\ = 25 \sqrt{2}-50 + 5 \sqrt{3}- 5 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 5 \sqrt{2}\right) } \cdot \left( 5- 5 \sqrt{2}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot5+\color{blue}{ 5 \sqrt{2}} \cdot- 5 \sqrt{2} = \\ = 25- 25 \sqrt{2} + 25 \sqrt{2}-50 $$ |
| 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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