If \(x+\displaystyle\frac{1}{x}=4\), then find the value of \(x^5+\displaystyle\frac{1}{x^5}\).
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 724
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.
\(x^2+\displaystyle\frac{1}{x^2}=4^2-2=14\)
\(x^3+\displaystyle\frac{1}{x^3}=4^3-3\times4=64-12=52\)
\(14\times52=x^5+\displaystyle\frac{1}{x^5}+4\), so \(728=x^5+\displaystyle\frac{1}{x^5}+4\)
\(x^5+\displaystyle\frac{1}{x^5}=728-4=724\)