Question
Answer
$$\dfrac{3\sqrt{2}}{\sqrt{18}-\sqrt{45}}=-\dfrac{2+\sqrt{10}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}}{\sqrt{18}-\sqrt{45}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}}{\sqrt{18}-\sqrt{45}}\frac{\sqrt{18}+\sqrt{45}}{\sqrt{18}+\sqrt{45}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18+9\sqrt{10}}{18+9\sqrt{10}-9\sqrt{10}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18+9\sqrt{10}}{-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2+\sqrt{10}}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{2+\sqrt{10}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{18} + \sqrt{45}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{2} } \cdot \left( \sqrt{18} + \sqrt{45}\right) = \color{blue}{ 3 \sqrt{2}} \cdot \sqrt{18}+\color{blue}{ 3 \sqrt{2}} \cdot \sqrt{45} = \\ = 18 + 9 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{18}- \sqrt{45}\right) } \cdot \left( \sqrt{18} + \sqrt{45}\right) = \color{blue}{ \sqrt{18}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot \sqrt{45}\color{blue}{- \sqrt{45}} \cdot \sqrt{18}\color{blue}{- \sqrt{45}} \cdot \sqrt{45} = \\ = 18 + 9 \sqrt{10}- 9 \sqrt{10}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Place a negative sign in front of a fraction. |
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