If \(x+\displaystyle\frac{1}{x}=6\), then find the value of \(x^5+\displaystyle\frac{1}{x^5}\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6726
The fifth power comes from multiplying the second and third powers: \(\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=x^5+\frac{1}{x^5}+x+\frac{1}{x}\). So the square and the cube are found first.
\(x^2+\displaystyle\frac{1}{x^2}=6^2-2=34\)
\(x^3+\displaystyle\frac{1}{x^3}=6^3-3\times6=216-18=198\)
\(34\times198=x^5+\displaystyle\frac{1}{x^5}+6\), so \(6732=x^5+\displaystyle\frac{1}{x^5}+6\)
\(x^5+\displaystyle\frac{1}{x^5}=6732-6=6726\)