Question
Answer
$$\dfrac{\sqrt{4}+5\sqrt{2}}{\sqrt{4}\cdot8+\sqrt{18}}=\dfrac{1+37\sqrt{2}}{119}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{4}+5\sqrt{2}}{\sqrt{4}\cdot8+\sqrt{18}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{4}+5\sqrt{2}}{\sqrt{4}\cdot8+\sqrt{18}}\frac{8\sqrt{4}-\sqrt{18}}{8\sqrt{4}-\sqrt{18}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{32-6\sqrt{2}+80\sqrt{2}-30}{256-48\sqrt{2}+48\sqrt{2}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2+74\sqrt{2}}{238} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+37\sqrt{2}}{119}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8 \sqrt{4}- \sqrt{18}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{4} + 5 \sqrt{2}\right) } \cdot \left( 8 \sqrt{4}- \sqrt{18}\right) = \color{blue}{ \sqrt{4}} \cdot 8 \sqrt{4}+\color{blue}{ \sqrt{4}} \cdot- \sqrt{18}+\color{blue}{ 5 \sqrt{2}} \cdot 8 \sqrt{4}+\color{blue}{ 5 \sqrt{2}} \cdot- \sqrt{18} = \\ = 32- 6 \sqrt{2} + 80 \sqrt{2}-30 $$ Simplify denominator. $$ \color{blue}{ \left( 8 \sqrt{4} + \sqrt{18}\right) } \cdot \left( 8 \sqrt{4}- \sqrt{18}\right) = \color{blue}{ 8 \sqrt{4}} \cdot 8 \sqrt{4}+\color{blue}{ 8 \sqrt{4}} \cdot- \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot 8 \sqrt{4}+\color{blue}{ \sqrt{18}} \cdot- \sqrt{18} = \\ = 256- 48 \sqrt{2} + 48 \sqrt{2}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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