Question : If $x+\frac{1}{x}=-6$, what will be the value of $x^5+\frac{1}{x^5}$?
Option 1: – 7776
Option 2: – 6726
Option 3: – 6738
Option 4: – 6732
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Correct Answer: – 6726
Solution :
Given:
$x+\frac{1}{x}=-6$ ---------------------------------------(1)
Squaring both sides of equation (1)
So, $(x+\frac{1}{x})^2=(-6)^2$
⇒ $x^2+\frac{1}{x^2}=36-2=34$
Cubing both sides of equation (1)
$(x+\frac{1}{x})^3=(-6)^3$
⇒ $x^3+\frac{1}{x^3}+3×x×\frac{1}{x}(x+\frac{1}{x})=-216$
⇒ $x^3+\frac{1}{x^3}+3(-6)=-216$
⇒ $x^3+\frac{1}{x^3}=-216+18=-198$
So, $x^5+\frac{1}{x^5}$
= $(x^2+\frac{1}{x^2})(x^3+\frac{1}{x^3})-(x+\frac{1}{x})$
= $34×(-198)-(-6)$
= $-6732+6$
= $-6726$
Hence, the correct answer is – 6726.
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