Question
Answer
$$\dfrac{45-16\sqrt{10}}{30\sqrt{10}+24}=\dfrac{289\sqrt{10}-980}{1404}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{45-16\sqrt{10}}{30\sqrt{10}+24}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{45-16\sqrt{10}}{30\sqrt{10}+24}\frac{30\sqrt{10}-24}{30\sqrt{10}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1350\sqrt{10}-1080-4800+384\sqrt{10}}{9000-720\sqrt{10}+720\sqrt{10}-576} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1734\sqrt{10}-5880}{8424} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{289\sqrt{10}-980}{1404}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 30 \sqrt{10}-24} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 45- 16 \sqrt{10}\right) } \cdot \left( 30 \sqrt{10}-24\right) = \color{blue}{45} \cdot 30 \sqrt{10}+\color{blue}{45} \cdot-24\color{blue}{- 16 \sqrt{10}} \cdot 30 \sqrt{10}\color{blue}{- 16 \sqrt{10}} \cdot-24 = \\ = 1350 \sqrt{10}-1080-4800 + 384 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 30 \sqrt{10} + 24\right) } \cdot \left( 30 \sqrt{10}-24\right) = \color{blue}{ 30 \sqrt{10}} \cdot 30 \sqrt{10}+\color{blue}{ 30 \sqrt{10}} \cdot-24+\color{blue}{24} \cdot 30 \sqrt{10}+\color{blue}{24} \cdot-24 = \\ = 9000- 720 \sqrt{10} + 720 \sqrt{10}-576 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
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