# How do you factor 216a^3b^3-343c^6?

##### 2 Answers
Jul 6, 2018

color(blue)((6ab-7c^2)(36a^2b^2+42abc^2+49c^4)

#### Explanation:

First notice:

$216 = {6}^{3}$

and:

$343 = {7}^{3}$

We can now write:

${6}^{3} {a}^{3} {b}^{3} - {7}^{3} {c}^{6}$

Which leads to:

${\left(6 a b\right)}^{3} - {\left(7 {c}^{2}\right)}^{3}$

This is the difference of two cubes:

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

$\therefore$

$\left(6 a b - 7 {c}^{2}\right) \left(36 {a}^{2} {b}^{2} + 42 a b {c}^{2} + 49 {c}^{4}\right)$

Jul 6, 2018

$\left(6 a b - 7 {c}^{2}\right) \left(36 {a}^{2} {b}^{2} + 42 a b {c}^{2} + 49 {c}^{4}\right)$

#### Explanation:

Note that

$216 {a}^{3} {b}^{3} = {\left(2 \cdot 3 \cdot a \cdot b\right)}^{3}$
and

$343 {c}^{6} = {\left(7 {c}^{2}\right)}^{2}$

we use the formula

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

$\left(6 a b - 7 {c}^{2}\right) \left(36 {a}^{2} {b}^{2} + 42 a b {c}^{2} + 49 {c}^{4}\right)$