Question

A = a^5+1/a^5 với a +1/a = 3

Tính A

Answer 1

a+1/a=3

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

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

\(\Leftrightarrow a^4+\dfrac{1}{a^4}+2=49\)

=>\(a^4+\dfrac{1}{a^4}=47\)

\(A=\left(a+\dfrac{1}{a}\right)\left(a^4-a^3\cdot\dfrac{1}{a}+a^2\cdot\dfrac{1}{a^2}-a\cdot\dfrac{1}{a^3}+\dfrac{1}{a^4}\right)\)

\(=3\left(a^4-a^2+1-\dfrac{1}{a^2}+\dfrac{1}{a^4}\right)\)

\(=3\left(47-7+1\right)=3\cdot41=123\)