Question

Factorise:

a⁶ – 64b⁶

Answer 1

We can factorise a⁶ – 64b⁶ as a difference of squares:

a⁶ – 64b⁶ = (a³)² – (8b³)²

Now, apply the difference of squares formula.

x² – y² = (x – y)(x + y):

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

Next, we can factor each of these terms further.

Notice that both a³ – 8b³ and a³ + 8b³ are special forms of cubes:

a³ – 8b³ = (a – 2b) (a² + 2ab + 4b²)

a³ + 8b³ = (a + 2b) (a² – 2ab + 4b²)

So the fully factorised form of a⁶ – 64b⁶ is a⁶ − 64b⁶ = (a − 2b) (a² + 2ab + 4b²) (a + 2b) (a² − 2ab + 4b²)