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Problem 234

AMC 10/12, early questions

Algebra Difficulty 3.9 Find the answer 16. tekmovanje v znanju matematike za dijake srednjih tehniških in strokovnih šol Državno tekmovanje · Slovenia

Problem:
Koliko je vrednost izraza $\left(\frac{\sqrt{5}-3}{2}\right)^{2016} \frac{\sqrt{5}-3}{2}\left(\frac{\sqrt{5}+3}{2}\right)^{2016}$ ?
(A) 1
(B) $\frac{\sqrt{5}}{2}$
(C) $\frac{\sqrt{5}-3}{2}$
(D) $\frac{3-\sqrt{5}}{2}$
(E) $\frac{(\sqrt{5})^{2016}+3^{2016}}{8}$

Your answer

Official solution

Solution:
$\begin{gathered} \left(\frac{\sqrt{5}-3}{2}\right)^{2016} \frac{\sqrt{5}-3}{2}\left(\frac{\sqrt{5}+3}{2}\right)^{2016}=\left(\frac{\sqrt{5}-3}{2} \cdot \frac{\sqrt{5}+3}{2}\right)^{2016} \frac{\sqrt{5}-3}{2}= \\ =\left(\frac{5-9}{4}\right)^{2016} \frac{\sqrt{5}-3}{2}=1 \cdot \frac{\sqrt{5}-3}{2}=\frac{\sqrt{5}-3}{2} \end{gathered}$

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