Simplify the expression: ((4.2)^3 - (1.2)^3) / ((4.2)^2 + 5.04 + (1.2)^2).

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Step 1: Recognize the numerator as a difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Here, a = 4.2 and b = 1.2.

Step 2: Calculate a - b = 4.2 - 1.2 = 3.0.

Step 3: Calculate the denominator: (4.2)^2 + 5.04 + (1.2)^2.
(4.2)^2 = 17.64, (1.2)^2 = 1.44.
Sum = 17.64 + 5.04 + 1.44 = 24.12.

Step 4: Calculate a^2 + ab + b^2 for the numerator factorization.
ab = 4.2 * 1.2 = 5.04.
Sum = 17.64 + 5.04 + 1.44 = 24.12.

Step 5: Substitute back:
Numerator = (a - b)(a^2 + ab + b^2) = 3 * 24.12 = 72.36.
Denominator = 24.12.

Step 6: Simplify the fraction:
72.36 / 24.12 = 3.

Therefore, the simplified value of the expression is 3.

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