is equal to

Concept: For any two real numbers a and b, the difference-of-cubes identity states a3 − b3 = (a − b)(a2 + ab + b2). Whenever an expression is exactly this cube-difference divided by that same quadratic (a2 + ab + b2), the shared factor cancels and the whole expression collapses to just a − b.

1. Application: From the image, the numerator is 8.9×8.9×8.9 − 3.7×3.7×3.7, i.e. a3 − b3 with a = 8.9 and b = 3.7; the denominator is 8.9×8.9 + 8.9×3.7 + 3.7×3.7, i.e. a2 + ab + b2 for the same a and b.

2. Factor the numerator using the identity: a3 − b3 = (a − b)(a2 + ab + b2).

3. The factor (a2 + ab + b2) is identical to the denominator, so it cancels, leaving just (a − b).

4. Substitute the values: a − b = 8.9 − 3.7 = 5.2.

Cross-check: Evaluate both parts directly without the identity — numerator = 8.9×8.9×8.9 − 3.7×3.7×3.7 = 704.969 − 50.653 = 654.316; denominator = 8.9×8.9 + 8.9×3.7 + 3.7×3.7 = 79.21 + 32.93 + 13.69 = 125.83; dividing, 654.316 ÷ 125.83 = 5.2, which matches the identity-based result.

Hence the expression equals 5.2.