Question
Answer
$$\dfrac{17\sqrt{19}}{\sqrt{19}-\sqrt{2}}=19+\sqrt{38}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{17\sqrt{19}}{\sqrt{19}-\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{17\sqrt{19}}{\sqrt{19}-\sqrt{2}}\frac{\sqrt{19}+\sqrt{2}}{\sqrt{19}+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{323+17\sqrt{38}}{19+\sqrt{38}-\sqrt{38}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{323+17\sqrt{38}}{17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{19+\sqrt{38}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}19+\sqrt{38}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{19} + \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 17 \sqrt{19} } \cdot \left( \sqrt{19} + \sqrt{2}\right) = \color{blue}{ 17 \sqrt{19}} \cdot \sqrt{19}+\color{blue}{ 17 \sqrt{19}} \cdot \sqrt{2} = \\ = 323 + 17 \sqrt{38} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{19}- \sqrt{2}\right) } \cdot \left( \sqrt{19} + \sqrt{2}\right) = \color{blue}{ \sqrt{19}} \cdot \sqrt{19}+\color{blue}{ \sqrt{19}} \cdot \sqrt{2}\color{blue}{- \sqrt{2}} \cdot \sqrt{19}\color{blue}{- \sqrt{2}} \cdot \sqrt{2} = \\ = 19 + \sqrt{38}- \sqrt{38}-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 17. |
| ⑤ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator