If x/y = (-1/3)⁻³ ÷ (2/3)⁻⁴, then what is the value of (x/y + y/x)⁻¹?

For a nonzero base, the negative-exponent law states a-n = 1/an, which for a fraction becomes (p/q)-n = (q/p)n. Once a fraction equals x/y, its reciprocal is y/x, and the sum of a fraction with its own reciprocal must be combined over a common denominator before that sum is inverted.

1. Simplify (-1/3)-3 using the negative-exponent law: (-1/3)-3 = (-3)3 = -27.

2. Simplify (2/3)-4: (2/3)-4 = (3/2)4 = 81/16.

3. Compute x/y = -27 ÷ (81/16) = -27 × 16/81 = -16/3.

4. Take the reciprocal: y/x = -3/16.

5. Add x/y + y/x over the common denominator 48: -16/3 - 3/16 = (-256 - 9)/48 = -265/48.

6. Invert the sum: (x/y + y/x)-1 = -48/265.

Cross-check with decimals: x/y ≈ -5.33 and y/x ≈ -0.19, so their sum is about -5.52; inverting gives about -0.181, which matches -48/265 ≈ -0.181.