Final answer:
To factor the expression 125m⁶ - 8n³, recognize it as a difference of cubes and use the formula a³ - b³ = (a - b)(a² + ab + b²) to get the factored form (5m² - 2n)(25m⁴ + 10m²n + 4n²).
Step-by-step explanation:
The expression 125m⁶ - 8n³ can be factored by recognizing it as a difference of cubes since 125 and 8 are both perfect cubes (125 = 5³ and 8 = 2³) and the variables are also raised to powers that are multiples of 3 (m⁶ = (m²)³ and n³ = n³).
Applying the difference of cubes formula, which is a³ - b³ = (a - b)(a² + ab + b²), we obtain:
Step 1: Identify a = 5m² and b = 2n. Therefore, our expression becomes (5m²)³ - (2n)³.
Step 2: Apply the difference of cubes formula to get:
(5m² - 2n)((5m²)² + (5m²)(2n) + (2n)²)
Step 3: Simplify the factors to get:
(5m² - 2n)(25m⁴ + 10m²n + 4n²)
This is the factored form of the original expression.