Question
Answer
$$\dfrac{4}{\sqrt{21}-6\sqrt{5}}=-\dfrac{4\sqrt{21}+24\sqrt{5}}{159}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{21}-6\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{21}-6\sqrt{5}}\frac{\sqrt{21}+6\sqrt{5}}{\sqrt{21}+6\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{21}+24\sqrt{5}}{21+6\sqrt{105}-6\sqrt{105}-180} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{21}+24\sqrt{5}}{-159} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{4\sqrt{21}+24\sqrt{5}}{159}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{21} + 6 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{21} + 6 \sqrt{5}\right) = \color{blue}{4} \cdot \sqrt{21}+\color{blue}{4} \cdot 6 \sqrt{5} = \\ = 4 \sqrt{21} + 24 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{21}- 6 \sqrt{5}\right) } \cdot \left( \sqrt{21} + 6 \sqrt{5}\right) = \color{blue}{ \sqrt{21}} \cdot \sqrt{21}+\color{blue}{ \sqrt{21}} \cdot 6 \sqrt{5}\color{blue}{- 6 \sqrt{5}} \cdot \sqrt{21}\color{blue}{- 6 \sqrt{5}} \cdot 6 \sqrt{5} = \\ = 21 + 6 \sqrt{105}- 6 \sqrt{105}-180 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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