Answer
$$\frac{5}{4\sqrt{2}+2\sqrt{3}}=\frac{2\sqrt{2}-\sqrt{3}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{4\sqrt{2}+2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{4\sqrt{2}+2\sqrt{3}}\frac{4\sqrt{2}-2\sqrt{3}}{4\sqrt{2}-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20\sqrt{2}-10\sqrt{3}}{32-8\sqrt{6}+8\sqrt{6}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20\sqrt{2}-10\sqrt{3}}{20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{2}-\sqrt{3}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{2}- 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 4 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{5} \cdot 4 \sqrt{2}+\color{blue}{5} \cdot- 2 \sqrt{3} = \\ = 20 \sqrt{2}- 10 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{2} + 2 \sqrt{3}\right) } \cdot \left( 4 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{ 4 \sqrt{2}} \cdot 4 \sqrt{2}+\color{blue}{ 4 \sqrt{2}} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot 4 \sqrt{2}+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 32- 8 \sqrt{6} + 8 \sqrt{6}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 10. |
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