Question
Answer
$$\dfrac{1}{3\sqrt{25}-10+5\sqrt{10}}=\dfrac{-1+\sqrt{10}}{45}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{3\sqrt{25}-10+5\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{15-10+5\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{5+5\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{5+5\sqrt{10}}\frac{5-5\sqrt{10}}{5-5\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5-5\sqrt{10}}{25-25\sqrt{10}+25\sqrt{10}-250} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{5-5\sqrt{10}}{-225} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{1-\sqrt{10}}{-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{-1+\sqrt{10}}{45}\end{aligned} $$ | |
| ① | $$ 3 \sqrt{25} = 3 \cdot 5 = 15 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 5 \sqrt{10}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 5- 5 \sqrt{10}\right) = \color{blue}{1} \cdot5+\color{blue}{1} \cdot- 5 \sqrt{10} = \\ = 5- 5 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 5 \sqrt{10}\right) } \cdot \left( 5- 5 \sqrt{10}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 5 \sqrt{10}+\color{blue}{ 5 \sqrt{10}} \cdot5+\color{blue}{ 5 \sqrt{10}} \cdot- 5 \sqrt{10} = \\ = 25- 25 \sqrt{10} + 25 \sqrt{10}-250 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Divide both numerator and denominator by 5. |
| ⑦ | Multiply both numerator and denominator by -1. |
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