Question
Answer
$$\dfrac{1975+8847\sqrt{5}}{1921+8847\sqrt{5}}=\dfrac{193776535+238869\sqrt{5}}{193828402}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1975+8847\sqrt{5}}{1921+8847\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1975+8847\sqrt{5}}{1921+8847\sqrt{5}}\frac{1921-8847\sqrt{5}}{1921-8847\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3793975-17472825\sqrt{5}+16995087\sqrt{5}-391347045}{3690241-16995087\sqrt{5}+16995087\sqrt{5}-391347045} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-387553070-477738\sqrt{5}}{-387656804} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-193776535-238869\sqrt{5}}{-193828402} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{193776535+238869\sqrt{5}}{193828402}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1921- 8847 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1975 + 8847 \sqrt{5}\right) } \cdot \left( 1921- 8847 \sqrt{5}\right) = \color{blue}{1975} \cdot1921+\color{blue}{1975} \cdot- 8847 \sqrt{5}+\color{blue}{ 8847 \sqrt{5}} \cdot1921+\color{blue}{ 8847 \sqrt{5}} \cdot- 8847 \sqrt{5} = \\ = 3793975- 17472825 \sqrt{5} + 16995087 \sqrt{5}-391347045 $$ Simplify denominator. $$ \color{blue}{ \left( 1921 + 8847 \sqrt{5}\right) } \cdot \left( 1921- 8847 \sqrt{5}\right) = \color{blue}{1921} \cdot1921+\color{blue}{1921} \cdot- 8847 \sqrt{5}+\color{blue}{ 8847 \sqrt{5}} \cdot1921+\color{blue}{ 8847 \sqrt{5}} \cdot- 8847 \sqrt{5} = \\ = 3690241- 16995087 \sqrt{5} + 16995087 \sqrt{5}-391347045 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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