Question : If $ x+\frac{1}{x}=3,$ then the value of $x^{5}+\frac{1}{x^{5}}$ is:
Option 1: 110
Option 2: 132
Option 3: 122
Option 4: 123
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Correct Answer: 123
Solution :
$x+\frac{1}{x }=3$.........................................(1)
Squaring both sides, we get,
$⇒(x+\frac{1}{x })^2=3^2$
$⇒x^2+\frac{1}{x^2 } +2=9$
$⇒x^2+\frac{1}{x^2 } =7$..................................(2)
Cubing both sides of equation (1), we get,
$⇒(x+\frac{1}{x})^3=x^3 + \frac{1}{x^3}+3(x+\frac{1}{x})$
$⇒3^3=x^3 + \frac{1}{x^3}+3(3)$
$⇒x^3 + \frac{1}{x^3}=18$..................................(3)
Multiplying (2) and (3), we get,
$⇒\left ( x^2+\frac{1}{x^2} \right )\cdot \left ( x^3+\frac{1}{x^3} \right )=7\times 18 = 126$
$⇒x^5+\frac{1}{x^5}+x+\frac{1}{x} = 126$
$⇒x^5+\frac{1}{x^5}+3 = 126$
$\therefore x^5+\frac{1}{x^5} = 123$
Hence, the correct answer is 123.
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