Question
Answer
$$\dfrac{1}{11\sqrt{5}-3\sqrt{3}}=\dfrac{11\sqrt{5}+3\sqrt{3}}{578}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{11\sqrt{5}-3\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{11\sqrt{5}-3\sqrt{3}}\frac{11\sqrt{5}+3\sqrt{3}}{11\sqrt{5}+3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11\sqrt{5}+3\sqrt{3}}{605+33\sqrt{15}-33\sqrt{15}-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{11\sqrt{5}+3\sqrt{3}}{578}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11 \sqrt{5} + 3 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 11 \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{1} \cdot 11 \sqrt{5}+\color{blue}{1} \cdot 3 \sqrt{3} = \\ = 11 \sqrt{5} + 3 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 11 \sqrt{5}- 3 \sqrt{3}\right) } \cdot \left( 11 \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{ 11 \sqrt{5}} \cdot 11 \sqrt{5}+\color{blue}{ 11 \sqrt{5}} \cdot 3 \sqrt{3}\color{blue}{- 3 \sqrt{3}} \cdot 11 \sqrt{5}\color{blue}{- 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 605 + 33 \sqrt{15}- 33 \sqrt{15}-27 $$ |
| ③ | Simplify numerator and denominator |
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