Factorize:$8a^3 - 27b^3 - 36a^2b + 54ab^2$

Given:

$8a^3 - 27b^3 - 36a^2b + 54ab^2$

To do:

We have to factorize the given expression.

Solution:

We know that,

$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$

Therefore,

$8a^3 - 27b^3 - 36a^2b + 54ab^2 = (2a)^3 - (3b)^3 - 3 \times (2a)^2 \times 3b + 3 \times 2a \times (3b)^2$

$= (2a - 3b)^3$

$= (2a - 3b) (2a - 3b) (2a - 3b)$

Hence, $8a^3 - 27b^3 - 36a^2b + 54ab^2 = (2a - 3b) (2a - 3b) (2a - 3b)$.