# How do you factor 27a^3-64b^3?

Jan 8, 2016

$27 {a}^{3} - 64 {b}^{3} = \left(3 a - 4 b\right) \left(9 {a}^{2} + 12 a b + 16 {b}^{2}\right)$

#### Explanation:

Remembering that:

${a}^{3} - {b}^{3} = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$

we can try to write

$27 {a}^{3} - 64 {b}^{3}$

like a difference of cubes

$27 {a}^{3} - 64 {b}^{3} = {3}^{3} {a}^{3} - {2}^{6} {b}^{3} = {3}^{3} {a}^{3} - {\left({2}^{2}\right)}^{3} {b}^{3} =$
${\left(3 a\right)}^{3} - {4}^{3} {b}^{3} = {\left(3 a\right)}^{3} - {\left(4 b\right)}^{3}$

Now we can apply the rule:

$27 {a}^{3} - 64 {b}^{3} = {\left(3 a\right)}^{3} - {\left(4 b\right)}^{3} =$
$= \left(3 a - 4 b\right) \left({\left(3 a\right)}^{2} + 12 a b + {\left(4 b\right)}^{2}\right)$
$= \left(3 a - 4 b\right) \left(9 {a}^{2} + 12 a b + 16 {b}^{2}\right)$