If (x² + 1)/x = 3, what is the value of (x¹² + 1)/x ⁶ ?
Given, (x² + 1)/x = 3 ⇒ (x + 1/x) = 3 ⇒ (x + 1/x)² = 3² [Squaring on both sides] ⇒ x² + 1/x² + 2 = 9 ⇒ x² + 1/x² = 7 ⇒ (x² + 1/x²)³ = 7³ [Cubing on both sides] ⇒ (x²)³ + (1/x²)³ + 3 .x² . 1/x² (x² + 1/x²) = 343 ⇒ x ⁶ + 1/x ⁶ + 3 (x² + 1/x²) = 343 ⇒ x ⁶ + 1/x ⁶ + 3 * 7 = 343 ⇒ x ⁶ + 1/x ⁶ = 322 ⇒ (x¹² + 1)/x ⁶ = 322
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