# How do you find the product (2r-3t)^3?

Feb 26, 2017

See the entire solution process below:

#### Explanation:

First, we can rewrite this as:

$\left(2 r - 3 t\right) \left(2 r - 3 t\right) \left(2 r - 3 t\right)$

We can then find the product of the first two terms. To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{2 r} - \textcolor{red}{3 t}\right) \left(\textcolor{b l u e}{2 r} - \textcolor{b l u e}{3 t}\right) \left(2 r - 3 t\right)$ becomes:

$\left(\left(\textcolor{red}{2 r} \times \textcolor{b l u e}{2 r}\right) - \left(\textcolor{red}{2 r} \times \textcolor{b l u e}{3 t}\right) - \left(\textcolor{red}{3 t} \times \textcolor{b l u e}{2 r}\right) + \left(\textcolor{red}{3 t} \times \textcolor{b l u e}{3 t}\right)\right) \left(2 r - 3 t\right)$

$\left(4 {r}^{2} - 6 r t - 6 r t + 9 {t}^{2}\right) \left(2 r - 3 t\right)$

We can now combine like terms:

$\left(4 {r}^{2} + \left(- 6 - 6\right) r t + 9 {t}^{2}\right) \left(2 r - 3 t\right)$

$\left(4 {r}^{2} - 12 r t + 9 {t}^{2}\right) \left(2 r - 3 t\right)$

We can now multiply these two terms. To multiply these two terms you again multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{4 {r}^{2}} - \textcolor{red}{12 r t} + \textcolor{red}{9 {t}^{2}}\right) \left(\textcolor{b l u e}{2 r} - \textcolor{b l u e}{3 t}\right)$ becomes:

$\left(\textcolor{red}{4 {r}^{2}} \times \textcolor{b l u e}{2 r}\right) - \left(\textcolor{red}{4 {r}^{2}} \times \textcolor{b l u e}{3 t}\right) - \left(\textcolor{red}{12 r t} \times \textcolor{b l u e}{2 r}\right) + \left(\textcolor{red}{12 r t} \times \textcolor{b l u e}{3 t}\right) + \left(\textcolor{red}{9 {t}^{2}} \times \textcolor{b l u e}{2 r}\right) - \left(\textcolor{red}{9 {t}^{2}} \times \textcolor{b l u e}{3 t}\right)$

$8 {r}^{3} - 12 {r}^{2} t - 24 {r}^{2} t + 36 r {t}^{2} + 18 r {t}^{2} - 27 {t}^{3}$

We can now combine like terms:

$8 {r}^{3} + \left(- 12 - 24\right) {r}^{2} t + \left(36 + 18\right) r {t}^{2} - 27 {t}^{3}$

$8 {r}^{3} - 36 {r}^{2} t + 54 r {t}^{2} - 27 {t}^{3}$