# How do you simplify (−3c−3w^5)^3?

Jul 18, 2017

See a solution process below:

#### Explanation:

We can use Pascal's Triangle to simplify this expression.

The triangle values for the exponent 3 are:

$\textcolor{red}{1} \textcolor{w h i t e}{\ldots \ldots \ldots} \textcolor{red}{3} \textcolor{w h i t e}{\ldots \ldots \ldots} \textcolor{red}{3} \textcolor{w h i t e}{\ldots \ldots \ldots} \textcolor{red}{1}$

We can also write ${\left(- 3 x - 3 {w}^{5}\right)}^{3}$ as ${\left(- 3 x + - 3 {w}^{5}\right)}^{3}$

Therefore ${\left(\textcolor{b l u e}{- 3 c} + \textcolor{g r e e n}{- 3 {w}^{5}}\right)}^{3}$ can be multiplied as:

$\textcolor{red}{1} \left({\textcolor{g r e e n}{\left(- 3 {w}^{5}\right)}}^{0} {\textcolor{b l u e}{\left(- 3 c\right)}}^{3}\right) + \textcolor{red}{3} \left({\textcolor{g r e e n}{\left(- 3 {w}^{5}\right)}}^{1} {\textcolor{b l u e}{\left(- 3 c\right)}}^{2}\right) + \textcolor{red}{3} \left({\textcolor{g r e e n}{\left(- 3 {w}^{5}\right)}}^{2} {\textcolor{b l u e}{\left(- 3 c\right)}}^{1}\right) + \textcolor{red}{1} \left({\textcolor{g r e e n}{\left(- 3 {w}^{5}\right)}}^{3} {\textcolor{b l u e}{\left(- 3 c\right)}}^{0}\right)$

(color(red)(1) * color(green)(1) * color(blue)(-27c^3)) + (color(red)(3) * color(green)(-3w^5) * color(blue)(9c^2)) + (color(red)(3) * color(green)(9w^10) * color(blue)(-3c)) + (color(red)(1) * color(green)(-27w^15) * 1)

$- 27 {c}^{3} + \left(- 81 {w}^{5} {c}^{2}\right) + \left(- 81 {w}^{10} c\right) + \left(- 27 {w}^{15}\right)$

$- 27 {c}^{3} - 81 {w}^{5} {c}^{2} - 81 {w}^{10} c - 27 {w}^{15}$

$- 27 {c}^{3} - 81 {c}^{2} {w}^{5} - 81 c {w}^{10} - 27 {w}^{15}$