Simplify (-3r^3w^4)^3(2rw)^2(-3r^2)^3(4rw^2)^3(2r^2w^3)

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 3 r^{3} w^{4}$ .

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

Apply the product rule to $- 3 r^{3}$ .

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

Step 3

Raise $- 3$ to the power of $3$ .

$2 (- 27 {(r^{3})}^{3} {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} (r^{2} w^{3})$

Step 4

Multiply the exponents in ${(r^{3})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (- 27 r^{3 \cdot 3} {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} (r^{2} w^{3})$

Multiply $3$ by $3$ .

$2 (- 27 r^{9} {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} (r^{2} w^{3})$

Step 5

Multiply $r^{9}$ by $r^{2}$ by adding the exponents.

Move $r^{2}$ .

$2 (- 27 (r^{2} r^{9}) {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} w^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (- 27 r^{2 + 9} {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} w^{3}$

Add $2$ and $9$ .

$2 (- 27 r^{11} {(w^{4})}^{3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} w^{3}$

Step 6

Multiply the exponents in ${(w^{4})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (- 27 r^{11} w^{4 \cdot 3}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} w^{3}$

Multiply $4$ by $3$ .

$2 (- 27 r^{11} w^{12}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3} w^{3}$

Step 7

Multiply $w^{12}$ by $w^{3}$ by adding the exponents.

Move $w^{3}$ .

$2 (- 27 r^{11} (w^{3} w^{12})) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (- 27 r^{11} w^{3 + 12}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Add $3$ and $12$ .

$2 (- 27 r^{11} w^{15}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 8

Multiply $- 27$ by $2$ .

$- 54 (r^{11} w^{15}) {(2 r w)}^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 r w$ .

$- 54 r^{11} w^{15} ({(2 r)}^{2} w^{2}) {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Apply the product rule to $2 r$ .

$- 54 r^{11} w^{15} (2^{2} r^{2} w^{2}) {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 10

Multiply $r^{11}$ by $r^{2}$ by adding the exponents.

Move $r^{2}$ .

$- 54 (r^{2} r^{11}) w^{15} (2^{2} w^{2}) {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 54 r^{2 + 11} w^{15} (2^{2} w^{2}) {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Add $2$ and $11$ .

$- 54 r^{13} w^{15} (2^{2} w^{2}) {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 11

Multiply $w^{15}$ by $w^{2}$ by adding the exponents.

Move $w^{2}$ .

$- 54 r^{13} (w^{2} w^{15}) \cdot 2^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 54 r^{13} w^{2 + 15} \cdot 2^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Add $2$ and $15$ .

$- 54 r^{13} w^{17} \cdot 2^{2} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 12

Raise $2$ to the power of $2$ .

$- 54 r^{13} w^{17} \cdot 4 {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 13

Multiply $4$ by $- 54$ .

$- 216 r^{13} w^{17} {(- 3 r^{2})}^{3} {(4 r w^{2})}^{3}$

Step 14

Apply the product rule to $- 3 r^{2}$ .

$- 216 r^{13} w^{17} ({(- 3)}^{3} {(r^{2})}^{3}) {(4 r w^{2})}^{3}$

Step 15

Raise $- 3$ to the power of $3$ .

$- 216 r^{13} w^{17} (- 27 {(r^{2})}^{3}) {(4 r w^{2})}^{3}$

Step 16

Multiply the exponents in ${(r^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 216 r^{13} w^{17} (- 27 r^{2 \cdot 3}) {(4 r w^{2})}^{3}$

Multiply $2$ by $3$ .

$- 216 r^{13} w^{17} (- 27 r^{6}) {(4 r w^{2})}^{3}$

Step 17

Multiply $r^{13}$ by $r^{6}$ by adding the exponents.

Move $r^{6}$ .

$- 216 (r^{6} r^{13}) w^{17} \cdot - 27 {(4 r w^{2})}^{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 216 r^{6 + 13} w^{17} \cdot - 27 {(4 r w^{2})}^{3}$

Add $6$ and $13$ .

$- 216 r^{19} w^{17} \cdot - 27 {(4 r w^{2})}^{3}$

Step 18

Multiply $- 27$ by $- 216$ .

$5832 r^{19} w^{17} {(4 r w^{2})}^{3}$

Step 19

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $4 r w^{2}$ .

$5832 r^{19} w^{17} ({(4 r)}^{3} {(w^{2})}^{3})$

Apply the product rule to $4 r$ .

$5832 r^{19} w^{17} (4^{3} r^{3} {(w^{2})}^{3})$

Step 20

Multiply $r^{19}$ by $r^{3}$ by adding the exponents.

Move $r^{3}$ .

$5832 (r^{3} r^{19}) w^{17} (4^{3} {(w^{2})}^{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5832 r^{3 + 19} w^{17} (4^{3} {(w^{2})}^{3})$

Add $3$ and $19$ .

$5832 r^{22} w^{17} (4^{3} {(w^{2})}^{3})$

Step 21

Raise $4$ to the power of $3$ .

$5832 r^{22} w^{17} (64 {(w^{2})}^{3})$

Step 22

Multiply the exponents in ${(w^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5832 r^{22} w^{17} (64 w^{2 \cdot 3})$

Multiply $2$ by $3$ .

$5832 r^{22} w^{17} (64 w^{6})$

Step 23

Multiply $w^{17}$ by $w^{6}$ by adding the exponents.

Move $w^{6}$ .

$5832 r^{22} (w^{6} w^{17}) \cdot 64$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5832 r^{22} w^{6 + 17} \cdot 64$

Add $6$ and $17$ .

$5832 r^{22} w^{23} \cdot 64$

Step 24

Multiply $64$ by $5832$ .

$373248 r^{22} w^{23}$

Simplify (-3r^3w^4)^3(2rw)^2(-3r^2)^3(4rw^2)^3(2r^2w^3)

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