Question

Factorise:

15625a⁶ – 64b⁶ 

Answer 1

Factorise 15625a⁶ – 64b⁶  stepwise, formal

Step 1: Write as perfect sixth powers

15625a⁶ – 64b⁶ = 5⁶a⁶ – 2⁶b⁶

= [(5a)³]² – [(2b)³]²

= (125a³)² – (8b³)²

Step 2: Use difference of squares X² – Y² = (X – Y) (X + Y) 

(125a³)² – (8b³)² = (125a³ – 8b³) (125a³ + 8b³)

Step 3: Factor each cubic (cube identities)

A³ – B³ = (A – B) (A² + AB + B²),

A³ + B³ = (A + B) (A² – AB + B²)

With A = 5a, B = 2b:

125a³ – 8b³ = (5a – 2b) (25a² + 10ab + 4b²),

125a³ + 8b³ = (5a + 2b) (25a² – 10ab + 4b²).

Step 4: Combine the factors

(5a – 2b) (25a² + 10ab + 4b²) (5a + 2b) (25a² – 10ab + 4b²)