Question
Answer
$$\dfrac{1}{5\sqrt{3}-2\sqrt{2}}=\dfrac{5\sqrt{3}+2\sqrt{2}}{67}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{5\sqrt{3}-2\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{5\sqrt{3}-2\sqrt{2}}\frac{5\sqrt{3}+2\sqrt{2}}{5\sqrt{3}+2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{3}+2\sqrt{2}}{75+10\sqrt{6}-10\sqrt{6}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{3}+2\sqrt{2}}{67}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3} + 2 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 5 \sqrt{3} + 2 \sqrt{2}\right) = \color{blue}{1} \cdot 5 \sqrt{3}+\color{blue}{1} \cdot 2 \sqrt{2} = \\ = 5 \sqrt{3} + 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3}- 2 \sqrt{2}\right) } \cdot \left( 5 \sqrt{3} + 2 \sqrt{2}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot 2 \sqrt{2}\color{blue}{- 2 \sqrt{2}} \cdot 5 \sqrt{3}\color{blue}{- 2 \sqrt{2}} \cdot 2 \sqrt{2} = \\ = 75 + 10 \sqrt{6}- 10 \sqrt{6}-8 $$ |
| ③ | Simplify numerator and denominator |
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