Ta có: \(x^2+\frac{1}{x^2}=7\)
=>\(x^2+\frac{1}{x^2}+2-2=7\)
=>\(\left(x+\frac{1}{x}\right)^2-2=7\)
=>\(\left(x+\frac{1}{x}\right)^2=9\)
=>\(x+\frac{1}{x}=3\)
\(x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3\cdot x\cdot\frac{1}{x}\left(x+\frac{1}{x}\right)\)
\(=3^3-3\cdot3=27-9=18\)
\(\left(x^3+\frac{1}{x^3}\right)\left(x^2+\frac{1}{x^2}\right)\)
\(=x^5+\frac{x^3}{x^2}+\frac{x^2}{x^3}+\frac{1}{x^5}\)
\(=x^5+\frac{1}{x^5}+\frac{1}{x}+x=x^5+\frac{1}{x^5}+3\)
=>\(7\cdot18=x^5+\frac{1}{x^5}+3\)
=>\(x^5+\frac{1}{x^5}=7\cdot18-3=126-3=123\)
=>\(x^5+\frac{1}{x^5}\) là một số nguyên
Số nguyên cần tìm là 123
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