Simplify [(4.2)³ − (1.2)³] ⁄ [(4.2)² + 5.04 + (1.2)²]

Correct — B. The constant 5.04 is not a stray number: 4.2 × 1.2 = 5.04.

So with a = 4.2 and b = 1.2 the fraction reads (a³ − b³) ⁄ (a² + ab + b²).

a³ − b³ = (a − b)(a² + ab + b²), so the bracket cancels.
Quotient = a − b = 4.2 − 1.2 = 3 → option (b)

Arithmetic check: (74.088 − 1.728) ⁄ (17.64 + 5.04 + 1.44) = 72.36 ⁄ 24.12 = 3.

Why the others are wrong

• (a)
  2 — The fraction collapses to a − b, and 4.2 − 1.2 = 3. Landing on 2 needs the two numbers to differ by 2, which happens only if 1.2 is misread as 2.2.
• (c)
  5 — 5 is a rounded a + b, the sum-of-cubes result. That identity needs the denominator a² − ab + b², but the middle term 5.04 here is added, not subtracted.
• (d)
  1 — 1 would mean the numerator and denominator are equal. They are 72.36 and 24.12, a ratio of exactly 3, because the cancelled bracket is the denominator itself.

(a³ − b³) ⁄ (a² + ab + b²) is a − b for any a and b. The question's only real work is recognising the ab term when it is printed as a decimal.

5.04 is the product 4.2 × 1.2, so the denominator matches the second factor of a³ − b³ and cancels against the numerator.

The companion result is (a³ + b³) ⁄ (a² − ab + b²) = a + b, and the only difference between the two is the sign in front of ab.

Key facts

• a³ − b³ = (a − b)(a² + ab + b²), so the quotient by that bracket is a − b.
• 4.2 × 1.2 = 5.04, which is the ab term in the denominator.
• The value of the expression is 72.36 ⁄ 24.12 = 3.

Common traps

• Expanding all four powers and losing a minute on decimal arithmetic.
• Applying the sum-of-cubes factorisation and answering a + b.
• Treating 5.04 as an unrelated constant and calling the denominator unfactorable.

No directly related past PYQ was found.