Question
Answer
$$\dfrac{1+1+1+1}{\sqrt{2}+1\sqrt{3}+\sqrt{2}\cdot\sqrt{4}+\sqrt{3}\cdot\sqrt{9}+\sqrt{8}}=\dfrac{20\sqrt{2}-16\sqrt{3}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+1+1+1}{\sqrt{2}+1\sqrt{3}+\sqrt{2}\cdot\sqrt{4}+\sqrt{3}\cdot\sqrt{9}+\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{2}+\sqrt{3}+2\sqrt{8}+\sqrt{27}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4}{\sqrt{2}+\sqrt{3}+4\sqrt{2}+3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4}{5\sqrt{2}+4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4}{5\sqrt{2}+4\sqrt{3}}\frac{5\sqrt{2}-4\sqrt{3}}{5\sqrt{2}-4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{20\sqrt{2}-16\sqrt{3}}{50-20\sqrt{6}+20\sqrt{6}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{20\sqrt{2}-16\sqrt{3}}{2}\end{aligned} $$ | |
| ① | Simplify numerator and denominator |
| ② | $$ 2 \sqrt{8} =
2 \sqrt{ 2 ^2 \cdot 2 } =
2 \sqrt{ 2 ^2 } \, \sqrt{ 2 } =
2 \cdot 2 \sqrt{ 2 } =
4 \sqrt{ 2 } $$ |
| ③ | $$ \sqrt{27} =
\sqrt{ 3 ^2 \cdot 3 } =
\sqrt{ 3 ^2 } \, \sqrt{ 3 } =
3 \sqrt{ 3 }$$ |
| ④ | Simplify numerator and denominator |
| ⑤ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{2}- 4 \sqrt{3}} $$. |
| ⑥ | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 5 \sqrt{2}- 4 \sqrt{3}\right) = \color{blue}{4} \cdot 5 \sqrt{2}+\color{blue}{4} \cdot- 4 \sqrt{3} = \\ = 20 \sqrt{2}- 16 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{2} + 4 \sqrt{3}\right) } \cdot \left( 5 \sqrt{2}- 4 \sqrt{3}\right) = \color{blue}{ 5 \sqrt{2}} \cdot 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot- 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot 5 \sqrt{2}+\color{blue}{ 4 \sqrt{3}} \cdot- 4 \sqrt{3} = \\ = 50- 20 \sqrt{6} + 20 \sqrt{6}-48 $$ |
| ⑦ | Simplify numerator and denominator |
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