Question : If $\frac{x^2-1}{x}=8$, then what is the value of $\frac{x^6-1}{x^3}=?$
Option 1: 496
Option 2: 536
Option 3: 512
Option 4: 488
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Correct Answer: 536
Solution :
Given: $\frac{(x^{2} - 1)}{x} = 8$
⇒ $x - \frac{1}{x} = 8$
Cubing on both sides,
⇒ $x^{3}- \frac{1}{x^{3}} - 3 \times x \times \frac{1}{x} \times (x - \frac{1}{x})= (8)^3$ [Since, $(a - b)^{3} = a^{3} - b^{3} - 3ab (a - b)$]
⇒ $x^{3}- \frac{1}{x^{3}} - 3 \times 8 = 512$
⇒ $x^{3} - \frac{1}{x^{3}} = 512+24$
⇒ $\frac{(x^{6} - 1)}{x^{3}} = 536$
Hence, the correct answer is 536.
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