Simplify (0.58 × 0.58 × 0.58 − 0.001) ⁄ (0.58 × 0.58 + 0.058 + 0.01).

Correct — C. The image asks you to simplify (0.58 × 0.58 × 0.58 − 0.001) ⁄ (0.58 × 0.58 + 0.058 + 0.01). That is the identity for a³ − b³ over a² + ab + b², with a = 0.58 and b = 0.1.

Numerator: 0.58³ − 0.001, and 0.1³ = 0.001, so it is a³ − b³
Denominator: 0.58² + 0.058 + 0.01, and ab = 0.58 × 0.1 = 0.058, b² = 0.01
a³ − b³ = (a − b)(a² + ab + b²), so the whole denominator cancels
Value = a − b = 0.58 − 0.1 = 0.48 → option (c)

Why the others are wrong

• (a)
  0.37 — 0.37 would require b = 0.21, but 0.21³ = 0.009261, nothing like the 0.001 the numerator actually subtracts.
• (b)
  0.28 — 0.28 would require b = 0.30, and 0.3³ = 0.027. The numerator subtracts 0.001, so b is 0.1.
• (d)
  0.42 — 0.42 would require b = 0.16. Only b = 0.1 makes the denominator's middle term equal the given 0.058.

The expression looks like heavy decimal arithmetic and is really one line of algebra. a³ − b³ = (a − b)(a² + ab + b²) — so dividing a³ − b³ by a² + ab + b² leaves a − b and nothing else.

Recognising it depends on spotting decimals as powers: 0.001 = 0.1³, 0.01 = 0.1², and the odd-looking 0.058 is 0.58 × 0.1.

The sign in the denominator's middle term is the tell. Plus ab goes with a³ − b³; minus ab goes with a³ + b³.

Key facts

• a³ − b³ = (a − b)(a² + ab + b²).
• a³ + b³ = (a + b)(a² − ab + b²) is the companion identity.
• 0.001 = 0.1³ and 0.01 = 0.1², so b = 0.1 throughout this expression.
• The middle term 0.058 is exactly ab = 0.58 × 0.1, which confirms the match before you compute anything.

Common traps

• Multiplying 0.58 × 0.58 × 0.58 out by hand and losing a decimal place.
• Reading b as 0.01 because 0.01 appears in the denominator, when b² = 0.01 means b = 0.1.
• Taking the denominator's +ab for −ab and reaching for the a³ + b³ identity instead.

No directly related past PYQ was found.