# How do you simplify (-3r^2w^4)^3(2rw)^2(-3r^2)^3(4rw^2)^3(2r^2w^3)^4?

Jan 9, 2018

${\left(- 3 {r}^{2} {w}^{4}\right)}^{3} {\left(2 r w\right)}^{2} {\left(- 3 {r}^{2}\right)}^{3} {\left(4 r {w}^{2}\right)}^{3} {\left(2 {r}^{2} {w}^{3}\right)}^{4} = 2985984 {r}^{25} {w}^{32}$

#### Explanation:

Simplify.

${\left(- 3 {r}^{2} {w}^{4}\right)}^{3} {\left(2 r w\right)}^{2} {\left(- 3 {r}^{2}\right)}^{3} {\left(4 r {w}^{2}\right)}^{3} {\left(2 {r}^{2} {w}^{3}\right)}^{4}$

Apply the exponent power rule: ${\left({a}^{m}\right)}^{n} = {a}^{m \cdot n}$

Simplify ${\left(- 3 {r}^{2} {w}^{4}\right)}^{3}$ to $\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right)$.

$\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right) {\left(2 r w\right)}^{2} {\left(- 3 {r}^{2}\right)}^{3} {\left(4 r {w}^{2}\right)}^{3} {\left(2 {r}^{2} {w}^{3}\right)}^{4}$

Simplify ${\left(2 r w\right)}^{2}$ to $\left({2}^{2} {r}^{2} {w}^{2}\right)$.

$\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right) \left({2}^{2} {r}^{2} {w}^{2}\right) {\left(- 3 {r}^{2}\right)}^{3} {\left(4 r {w}^{2}\right)}^{3} {\left(2 {r}^{2} {w}^{3}\right)}^{4}$

Simplilfy ${\left(- 3 {r}^{2}\right)}^{3}$ to $\left(- {3}^{3} {r}^{2 \cdot 3}\right)$.

$\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right) \left({2}^{2} {r}^{2} {w}^{2}\right) \left(- {3}^{3} {r}^{2 \cdot 3}\right) {\left(4 r {w}^{2}\right)}^{3} {\left(2 {r}^{2} {w}^{3}\right)}^{4}$

Simplify ${\left(4 r {w}^{2}\right)}^{3}$ to $\left({4}^{3} {r}^{3} {w}^{4 \cdot 3}\right)$.

$\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right) \left({2}^{2} {r}^{2} {w}^{2}\right) \left(- {3}^{3} {r}^{2 \cdot 3}\right) \left({4}^{3} {r}^{3} {w}^{4 \cdot 3}\right) {\left(2 {r}^{2} {w}^{3}\right)}^{4}$

Simplify ${\left(2 {r}^{2} {w}^{3}\right)}^{4}$ to $\left({2}^{4} {w}^{3 \cdot 4}\right)$.

$\left(- {3}^{3} {r}^{2 \cdot 3} {w}^{4 \cdot 3}\right) \left({2}^{2} {r}^{2} {w}^{2}\right) \left(- {3}^{3} {r}^{2 \cdot 3}\right) \left({4}^{3} {r}^{3} {w}^{2 \cdot 3}\right) \left({2}^{4} {r}^{2 \cdot 4} {w}^{3 \cdot 4}\right)$

Simplify.

$\left(- 27 {r}^{6} {w}^{12}\right) \left(4 {r}^{2} {w}^{2}\right) \left(- 27 {r}^{6}\right) \left(64 {r}^{3} {w}^{6}\right) \left(16 {r}^{8} {w}^{12}\right)$

Take out the constants.

$- 27 \times 4 \times - 27 \times 64 \times 16 \left({r}^{6} {w}^{12} {r}^{2} {w}^{2} {r}^{6} {r}^{3} {w}^{6} {r}^{8} {w}^{12}\right)$

Simplify $- 27 \times 4 \times - 27 \times 64 \times 16$ to $2985984$.

$2985984 \left({r}^{6} {w}^{12} {r}^{2} {w}^{2} {r}^{6} {r}^{3} {w}^{6} {r}^{8} {w}^{12}\right)$

Collect like variables.

$2985984 \left({r}^{6} {r}^{2} {r}^{6} {r}^{3} {r}^{8} {w}^{12} {w}^{2} {w}^{6} {w}^{12}\right)$

Apply exponent product rule: ${a}^{m} {a}^{n} = {a}^{a + n}$

$2985984 \left({r}^{6 + 2 + 6 + 3 + 8} {w}^{12 + 2 + 6 + 12}\right)$

Simplify.

$2985984 {r}^{25} {w}^{32}$