Question

x+1/x=3

x^5+1/x^5=?

Answer 1

Có: \(x+\frac{1}{x}=3\Rightarrow\left(x+\frac{1}{x}\right)^2=9\)

\(\Leftrightarrow x^2+\frac{1}{x^2}+2.x.\frac{1}{x}=9\Leftrightarrow x^2+\frac{1}{x^2}=7\)

Có: \(x+\frac{1}{x}=3\Rightarrow\left(x+\frac{1}{x}\right)^3=27\)

\(\Leftrightarrow x^3+\frac{1}{x^3}+3.x.\frac{1}{x}\left(x+\frac{1}{x}\right)=27\)

\(\Leftrightarrow x^3+\frac{1}{x^3}+9=27\)

\(\Leftrightarrow x^3+\frac{1}{x^3}=18\)

Suy ra: \(\left(x^2+\frac{1}{x^2}\right).\left(x^3+\frac{1}{x^3}\right)=18.7\)

\(\Leftrightarrow x^5+\frac{1}{x^5}+x+\frac{1}{x}=126\)

\(\Leftrightarrow x^5+\frac{1}{x^5}=126-3=123\)

OK!!