If x + (1/x) = 8, then find the value of x³ + (1/x³) _____?

488

512

500

None of these

Given:

x + (1/x) = 8

Cube both sides:

(x + (1/x))³ = 8³

Expand the left side:

x³ + 3x²(1/x) + 3x(1/x)² + (1/x)³ = 512

Simplify:

x³ + 3x + 3(1/x) + (1/x)³ = 512

Rearrange:

x³ + (1/x)³ + 3(x + (1/x)) = 512

Substitute x + (1/x) = 8:

x³ + (1/x)³ + 3(8) = 512

x³ + (1/x)³ + 24 = 512

Subtract 24 from both sides:

x³ + (1/x)³ = 488