If x⁴ + x²y² + y⁴ = 29 and x² - xy + y² = 5, then what is the value of 5xy?

We use the standard algebraic identity:

x⁴ + x²y² + y⁴ = (x² + xy + y²)(x² − xy + y²).

We are given x⁴ + x²y² + y⁴ = 29 and x² − xy + y² = 5.

Substituting, 29 = (x² + xy + y²) × 5, so x² + xy + y² = 29/5 = 5.8.

Now subtract the two symmetric expressions: (x² + xy + y²) − (x² − xy + y²) = 2xy.

So 2xy = 5.8 − 5 = 0.8, which gives xy = 0.4.

Therefore 5xy = 5 × 0.4 = 2.

Hence the value of 5xy is 2.