# How do you multiply (mn - 5)(mn - 5)(mn - 5)?

Apr 22, 2015

If you know the identity ${\left(a - b\right)}^{3}$,
then you don't have to multiply.

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

${\left(m n - 5\right)}^{3} = {m}^{3} \cdot {n}^{3} - 15 {m}^{2} \cdot {n}^{2} + 75 m n - 125$

If you don't remember the Identity, then you may multiply:
${\left(m n - 5\right)}^{2} \cdot \left(m n - 5\right)$

$= \left({m}^{2} {n}^{2} - 10 m n + 25\right) \left(m n - 5\right)$

$= {m}^{3} \cdot {n}^{3} - 15 {m}^{2} \cdot {n}^{2} + 75 \cdot m n - 125$