Question

\(\left(\sqrt{5}+2\right)^{x-1}=\left(\sqrt{5}-2\right)^{\dfrac{x-1}{x+1}}\)

Answer 1

\(\left(\sqrt{5}+2\right)^{x-1}=\left(\sqrt{5}-2\right)^{\dfrac{x-1}{x+1}}\)

=>\(\left(\sqrt{5}+2\right)^{x-1}=\left(\sqrt{5}+2\right)^{\dfrac{-x+1}{x+1}}\)

=>\(x-1=\dfrac{-x+1}{x+1}\)

=>\(\left(x-1\right)\left(1+\dfrac{1}{x+1}\right)=0\)

=>\(\left[{}\begin{matrix}x-1=0\\x+1=-1\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x=1\\x=-2\end{matrix}\right.\)