If x⁴ - 79x² + 1 = 0, then a value of x³ + x⁻³ can be:

Concept: For an expression built from x and 1/x, the standard technique is to set y = x + 1/x and build higher symmetric power-sums from it, since y2 = x2 + 1/x2 + 2 and y3 = x3 + 1/x3 + 3y. Dividing the given quartic by the right power of x reduces it to an equation purely in x2 + 1/x2, which is exactly what this identity chain needs.

Applying it here:

1. Divide x4 - 79x2 + 1 = 0 by x2 (valid since x = 0 does not satisfy the equation) to get x2 + 1/x2 = 79.

2. Use y2 = x2 + 1/x2 + 2 with y = x + 1/x: y2 = 79 + 2 = 81, so y = x + 1/x = ±9.

3. Use y3 = x3 + 1/x3 + 3y, i.e. x3 + 1/x3 = y3 - 3y.

4. Substitute y = 9: x3 + 1/x3 = 93 - 3(9) = 729 - 27 = 702. (Substituting y = -9 gives -702.)

Cross-check: Solving x4 - 79x2 + 1 = 0 as a quadratic in x2 gives x2 = (79 ± √6237)/2 ≈ 78.987 or 0.0127; taking x ≈ 8.888 gives x3 + 1/x3 ≈ 702, confirming the algebraic result.

Result: Of the two possible values ±702, only 702 is offered among the options, so 702 is the value x3 + x-3 can take.