Step 1: Find the value of x + 1/x
Given equation: x⁴ - x² + 1 = 0
Dividing both sides by x²:
(x⁴ / x²) - (x² / x²) + (1 / x²) = 0
x² - 1 + 1/x² = 0
x² + 1/x² = 1

Using the formula a² + b² = (a + b)² - 2ab:
(x + 1/x)² - 2(x)(1/x) = 1
(x + 1/x)² - 2 = 1
(x + 1/x)² = 3
x + 1/x = √3

Step 2: Find the value of x³ + 1/x³
Using the formula a³ + b³ = (a + b)³ - 3ab(a + b):
x³ + 1/x³ = (x + 1/x)³ - 3(x)(1/x)(x + 1/x)
Substituting (x + 1/x) = √3:
x³ + 1/x³ = (√3)³ - 3(√3)
x³ + 1/x³ = 3√3 - 3√3
x³ + 1/x³ = 0

Answer: 0