Question
Answer
$$\dfrac{\sqrt{11}-\sqrt{5}}{5\sqrt{2}+2\sqrt{3}}=\dfrac{5\sqrt{22}-2\sqrt{33}-5\sqrt{10}+2\sqrt{15}}{38}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{11}-\sqrt{5}}{5\sqrt{2}+2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}-\sqrt{5}}{5\sqrt{2}+2\sqrt{3}}\frac{5\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{22}-2\sqrt{33}-5\sqrt{10}+2\sqrt{15}}{50-10\sqrt{6}+10\sqrt{6}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{22}-2\sqrt{33}-5\sqrt{10}+2\sqrt{15}}{38}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{2}- 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11}- \sqrt{5}\right) } \cdot \left( 5 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{ \sqrt{11}} \cdot 5 \sqrt{2}+\color{blue}{ \sqrt{11}} \cdot- 2 \sqrt{3}\color{blue}{- \sqrt{5}} \cdot 5 \sqrt{2}\color{blue}{- \sqrt{5}} \cdot- 2 \sqrt{3} = \\ = 5 \sqrt{22}- 2 \sqrt{33}- 5 \sqrt{10} + 2 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{2} + 2 \sqrt{3}\right) } \cdot \left( 5 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{ 5 \sqrt{2}} \cdot 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot 5 \sqrt{2}+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 50- 10 \sqrt{6} + 10 \sqrt{6}-12 $$ |
| ③ | Simplify numerator and denominator |
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