The value of [(0.867)3 − (0.321)3] ÷ [(0.867)2 + 0.867 × 0.321 + (0.321)2] is:

Concept: For any two numbers a and b, the difference of cubes factorises as a3 − b3 = (a − b)(a2 + ab + b2). So whenever a2 + ab + b2 is not zero, the quotient (a3 − b3) ÷ (a2 + ab + b2) reduces to the single difference a − b, and the cubes never have to be evaluated.

Application

1. Match the expression to the identity: here a = 0.867 and b = 0.321, the numerator is a3 − b3, and the denominator is exactly a2 + ab + b2.

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

3. Cancel the common factor a2 + ab + b2, which is positive here, so the cancellation is valid; the expression becomes simply a − b.

4. Substitute the given values: a − b = 0.867 − 0.321 = 0.546.

Cross-check by direct computation

• Numerator: 0.8673 − 0.3213 = 0.651714363 − 0.033076161 = 0.618638202.

• Denominator: 0.8672 + 0.867 × 0.321 + 0.3212 = 0.751689 + 0.278307 + 0.103041 = 1.133037.

• Quotient: 0.618638202 ÷ 1.133037 = 0.546, which matches the factorised result exactly.

Watch out: If the numerator carried a plus sign, the sum-of-cubes identity a3 + b3 = (a + b)(a2 − ab + b2) would apply instead, and the middle term of the quadratic factor would be −ab, giving a + b. Mixing the two forms breaks the cancellation, so always check the sign pattern of the middle term before cancelling.