\(x+\frac{1}{x}=3\Rightarrow\left(x+\frac{1}{x}\right)^2=9\Leftrightarrow x^2+2+\frac{1}{x^2}=9\Rightarrow x^2+\frac{1}{x^2}=7\)\(\left(x+\frac{1}{x}\right)^3=27\Leftrightarrow x^3+\frac{1}{x^3}+3x+\frac{3}{x}=27\Leftrightarrow x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)=27\Leftrightarrow x^3+\frac{1}{x^3}+3.3=27\Rightarrow x^3+\frac{1}{x^3}=18\)\(\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=7.18=126\)
\(\Leftrightarrow x^5+\frac{1}{x^5}+x+\frac{1}{x}=126\Rightarrow x^5+\frac{1}{x^5}=126-3=123\)
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