Question
Answer
$$\dfrac{7+3\sqrt{5}-(7-3\sqrt{5})}{3+\sqrt{5}-(3-\sqrt{5})}=3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7+3\sqrt{5}-(7-3\sqrt{5})}{3+\sqrt{5}-(3-\sqrt{5})}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7+3\sqrt{5}-7+3\sqrt{5}}{3+\sqrt{5}-3+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{5}}{2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6\sqrt{5}}{2\sqrt{5}}\frac{\sqrt{5}}{\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{30}{10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}} \frac{ 30 : \color{orangered}{ 10 } }{ 10 : \color{orangered}{ 10 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{3}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}3\end{aligned} $$ | |
| ① | Remove the parentheses by changing the sign of each term within them. $$ -\left( 7- 3 \sqrt{5} \right) = -7 + 3 \sqrt{5} $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ -\left( 3- \sqrt{5} \right) = -3 + \sqrt{5} $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}} $$. |
| ⑤ | Multiply in a numerator. $$ \color{blue}{ 6 \sqrt{5} } \cdot \sqrt{5} = 30 $$ Simplify denominator. $$ \color{blue}{ 2 \sqrt{5} } \cdot \sqrt{5} = 10 $$ |
| ⑥ | Divide both the top and bottom numbers by $ \color{orangered}{ 10 } $. |
| ⑦ | Remove 1 from denominator. |
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