Question : If $x-\frac{1}{x}=-6$, what will be the value of $x^5-\frac{1}{x^5}?$
Option 1: –8898
Option 2: –8896
Option 3: –8886
Option 4: –8892
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Correct Answer: –8886
Solution :
Given: $x-\frac{1}{x}=-6$ ----------------(equation1)
Squaring both sides of equation 1, we get:
⇒ $x^2+\frac{1}{x^2}-2=(-6)^2$
⇒ $x^2+\frac{1}{x^2}=38$
Cubing both sides of equation 1, we get:
⇒ $x^3-\frac{1}{x^3}-3×x×\frac{1}{x}(x-\frac{1}{x})=(-6)^3$
⇒ $x^3-\frac{1}{x^3}=-216+3×(-6)$
⇒ $x^3-\frac{1}{x^3}=-234$
Now, $(x^2+\frac{1}{x^2})(x^3-\frac{1}{x^3})=38×(-234)$
⇒ $(x^5-\frac{1}{x^5})+(x-\frac{1}{x})=(-8892)$
⇒ $(x^5-\frac{1}{x^5})+(-6)=(-8892)$
$\therefore(x^5-\frac{1}{x^5})= -8886$
Hence, the correct answer is –8886.
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