Question

cho a+\(\dfrac{1}{a}\)=3 tính A=a\(^5\)+\(\dfrac{1}{a^5}\)

Answer 1

\(a+\dfrac{1}{a}=3\)

\(\Rightarrow\left(a+\dfrac{1}{a}\right)^2=9\)

\(\Rightarrow a^2+2+\dfrac{1}{a^2}=9\)

\(\Rightarrow a^2+\dfrac{1}{a^2}=7\)

\(a+\dfrac{1}{a}=3\)

\(\Rightarrow\left(a+\dfrac{1}{a}\right)^3=27\)

\(\Rightarrow a^3+\dfrac{1}{a^3}+3a.\dfrac{1}{a}\left(a+\dfrac{1}{a}\right)=27\)

\(\Rightarrow a^3+\dfrac{1}{a^3}+3.3=27\)

\(\Rightarrow a^3+\dfrac{1}{a^3}=18\)

\(\left(a^2+\dfrac{1}{a^2}\right)\left(a^3+\dfrac{1}{a^3}\right)=7.18=126\)

\(\Rightarrow a^5+\dfrac{1}{a}+a+\dfrac{1}{a^5}=126\)

\(\Rightarrow a^5+3+\dfrac{1}{a^5}=126\)

\(\Rightarrow a^5+\dfrac{1}{a^5}=123\)