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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