Question
Answer
$$\dfrac{13\sqrt{2}}{\sqrt{18}-\sqrt{5}}=6+\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{13\sqrt{2}}{\sqrt{18}-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{13\sqrt{2}}{\sqrt{18}-\sqrt{5}}\frac{\sqrt{18}+\sqrt{5}}{\sqrt{18}+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{78+13\sqrt{10}}{18+3\sqrt{10}-3\sqrt{10}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{78+13\sqrt{10}}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6+\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}6+\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{18} + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 13 \sqrt{2} } \cdot \left( \sqrt{18} + \sqrt{5}\right) = \color{blue}{ 13 \sqrt{2}} \cdot \sqrt{18}+\color{blue}{ 13 \sqrt{2}} \cdot \sqrt{5} = \\ = 78 + 13 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{18}- \sqrt{5}\right) } \cdot \left( \sqrt{18} + \sqrt{5}\right) = \color{blue}{ \sqrt{18}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot \sqrt{18}\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 18 + 3 \sqrt{10}- 3 \sqrt{10}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 13. |
| ⑤ | Remove 1 from denominator. |
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