Question
Answer
$$\dfrac{4+3\sqrt{5}}{4+3\sqrt{5}}=1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+3\sqrt{5}}{4+3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+3\sqrt{5}}{4+3\sqrt{5}}\frac{4-3\sqrt{5}}{4-3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16-12\sqrt{5}+12\sqrt{5}-45}{16-12\sqrt{5}+12\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-29}{-29} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{29}{29} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}} \frac{ 29 : \color{orangered}{ 29 } }{ 29 : \color{orangered}{ 29 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}1\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + 3 \sqrt{5}\right) } \cdot \left( 4- 3 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot4+\color{blue}{ 3 \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 16- 12 \sqrt{5} + 12 \sqrt{5}-45 $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 3 \sqrt{5}\right) } \cdot \left( 4- 3 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot4+\color{blue}{ 3 \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 16- 12 \sqrt{5} + 12 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | cancel two minus signs |
| ⑤ | Divide both the top and bottom numbers by $ \color{orangered}{ 29 } $. |
| ⑥ | Remove 1 from denominator. |
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