Question
Answer
$$\dfrac{\sqrt{32}-\sqrt{20}}{\sqrt{24}+\sqrt{40}}=\dfrac{-4\sqrt{3}+4\sqrt{5}+\sqrt{30}-5\sqrt{2}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{32}-\sqrt{20}}{\sqrt{24}+\sqrt{40}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{32}-\sqrt{20}}{\sqrt{24}+\sqrt{40}}\frac{\sqrt{24}-\sqrt{40}}{\sqrt{24}-\sqrt{40}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16\sqrt{3}-16\sqrt{5}-4\sqrt{30}+20\sqrt{2}}{24-8\sqrt{15}+8\sqrt{15}-40} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16\sqrt{3}-16\sqrt{5}-4\sqrt{30}+20\sqrt{2}}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{3}-4\sqrt{5}-\sqrt{30}+5\sqrt{2}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-4\sqrt{3}+4\sqrt{5}+\sqrt{30}-5\sqrt{2}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{24}- \sqrt{40}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{32}- \sqrt{20}\right) } \cdot \left( \sqrt{24}- \sqrt{40}\right) = \color{blue}{ \sqrt{32}} \cdot \sqrt{24}+\color{blue}{ \sqrt{32}} \cdot- \sqrt{40}\color{blue}{- \sqrt{20}} \cdot \sqrt{24}\color{blue}{- \sqrt{20}} \cdot- \sqrt{40} = \\ = 16 \sqrt{3}- 16 \sqrt{5}- 4 \sqrt{30} + 20 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{24} + \sqrt{40}\right) } \cdot \left( \sqrt{24}- \sqrt{40}\right) = \color{blue}{ \sqrt{24}} \cdot \sqrt{24}+\color{blue}{ \sqrt{24}} \cdot- \sqrt{40}+\color{blue}{ \sqrt{40}} \cdot \sqrt{24}+\color{blue}{ \sqrt{40}} \cdot- \sqrt{40} = \\ = 24- 8 \sqrt{15} + 8 \sqrt{15}-40 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Multiply both numerator and denominator by -1. |
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