Find the value of the given expression, correct to 2 decimal places.…

Concept

For any numbers a and b, the sum-of-cubes identity is a3 + b3 = (a + b)(a2 − ab + b2).

Therefore, when the denominator is a2 − ab + b2 and is non-zero, the fraction simplifies to a + b.

Application

1. Let a = 0.05 and b = 0.04. The numerator is a3 + b3, and the denominator is a2 − ab + b2.

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

3. Evaluate the denominator: 0.0025 − 0.0020 + 0.0016 = 0.0021, so it is non-zero.

4. Cancel the common non-zero factor: a + b = 0.05 + 0.04 = 0.09.

Cross-check

1. The numerator is 0.000125 + 0.000064 = 0.000189.

2. The denominator is 0.0025 − 0.0020 + 0.0016 = 0.0021.

3. Direct division gives 0.000189 ÷ 0.0021 = 0.09, already expressed to two decimal places.