Question
Answer
$$\dfrac{5}{5-\sqrt{21}}=\dfrac{25+5\sqrt{21}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{5-\sqrt{21}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{5-\sqrt{21}}\frac{5+\sqrt{21}}{5+\sqrt{21}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25+5\sqrt{21}}{25+5\sqrt{21}-5\sqrt{21}-21} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25+5\sqrt{21}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + \sqrt{21}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 5 + \sqrt{21}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{21} = \\ = 25 + 5 \sqrt{21} $$ Simplify denominator. $$ \color{blue}{ \left( 5- \sqrt{21}\right) } \cdot \left( 5 + \sqrt{21}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{21}\color{blue}{- \sqrt{21}} \cdot5\color{blue}{- \sqrt{21}} \cdot \sqrt{21} = \\ = 25 + 5 \sqrt{21}- 5 \sqrt{21}-21 $$ |
| ③ | Simplify numerator and denominator |
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