Evaluate: (1.672 − 1.452)(1.67 − 1.45)

Concept: For any two numbers a and b, the difference of squares identity gives a2 − b2 = (a − b)(a + b). This turns a square-minus-square into a product of a sum and a difference, which is far easier to evaluate with decimals.

Application: Here a = 1.67 and b = 1.45, so the expression (1.672 − 1.452)(1.67 − 1.45) becomes (a − b)(a + b)(a − b), i.e. (a − b)2(a + b).

1. Compute a − b = 1.67 − 1.45 = 0.22.

2. Compute a + b = 1.67 + 1.45 = 3.12.

3. Factor the first bracket: 1.672 − 1.452 = (a − b)(a + b) = 0.22 × 3.12 = 0.6864.

4. Multiply by the remaining factor (a − b) = 0.22: 0.6864 × 0.22 = 0.151008 ≈ 0.151.

Cross-check: Computing directly without factoring gives the same result: 1.672 = 2.7889 and 1.452 = 2.1025, so 1.672 − 1.452 = 0.6864; multiplying by (1.67 − 1.45) = 0.22 again gives 0.6864 × 0.22 = 0.151008, which rounds to 0.151, confirming the factored calculation.