Question
Answer
$$\dfrac{5\sqrt{6}}{5+7\sqrt{21}}=\dfrac{-25\sqrt{6}+105\sqrt{14}}{1004}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{6}}{5+7\sqrt{21}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{6}}{5+7\sqrt{21}}\frac{5-7\sqrt{21}}{5-7\sqrt{21}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25\sqrt{6}-105\sqrt{14}}{25-35\sqrt{21}+35\sqrt{21}-1029} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25\sqrt{6}-105\sqrt{14}}{-1004} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-25\sqrt{6}+105\sqrt{14}}{1004}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 7 \sqrt{21}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 \sqrt{6} } \cdot \left( 5- 7 \sqrt{21}\right) = \color{blue}{ 5 \sqrt{6}} \cdot5+\color{blue}{ 5 \sqrt{6}} \cdot- 7 \sqrt{21} = \\ = 25 \sqrt{6}- 105 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 7 \sqrt{21}\right) } \cdot \left( 5- 7 \sqrt{21}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 7 \sqrt{21}+\color{blue}{ 7 \sqrt{21}} \cdot5+\color{blue}{ 7 \sqrt{21}} \cdot- 7 \sqrt{21} = \\ = 25- 35 \sqrt{21} + 35 \sqrt{21}-1029 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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