Simplify (-3a^3n^4)(-3a^3n)^4

$(- 3 a^{3} n^{4}) {(- 3 a^{3} n)}^{4}$

Step 1

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

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

$- 3 a^{3} n^{4} ({(- 3 a^{3})}^{4} n^{4})$

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

$- 3 a^{3} n^{4} ({(- 3)}^{4} {(a^{3})}^{4} n^{4})$

Step 2

Multiply $- 3$ by ${(- 3)}^{4}$ by adding the exponents.

Move ${(- 3)}^{4}$ .

${(- 3)}^{4} \cdot - 3 a^{3} n^{4} ({(a^{3})}^{4} n^{4})$

Multiply ${(- 3)}^{4}$ by $- 3$ .

Raise $- 3$ to the power of $1$ .

${(- 3)}^{4} \cdot {(- 3)}^{1} a^{3} n^{4} ({(a^{3})}^{4} n^{4})$

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

${(- 3)}^{4 + 1} a^{3} n^{4} ({(a^{3})}^{4} n^{4})$

Add $4$ and $1$ .

${(- 3)}^{5} a^{3} n^{4} ({(a^{3})}^{4} n^{4})$

Step 3

Multiply $n^{4}$ by $n^{4}$ by adding the exponents.

Move $n^{4}$ .

${(- 3)}^{5} a^{3} (n^{4} n^{4}) {(a^{3})}^{4}$

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

${(- 3)}^{5} a^{3} n^{4 + 4} {(a^{3})}^{4}$

Add $4$ and $4$ .

${(- 3)}^{5} a^{3} n^{8} {(a^{3})}^{4}$

Step 4

Raise $- 3$ to the power of $5$ .

$- 243 a^{3} n^{8} {(a^{3})}^{4}$

Step 5

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

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

$- 243 a^{3} n^{8} a^{3 \cdot 4}$

Multiply $3$ by $4$ .

$- 243 a^{3} n^{8} a^{12}$

Step 6

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

Move $a^{12}$ .

$- 243 (a^{12} a^{3}) n^{8}$

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

$- 243 a^{12 + 3} n^{8}$

Add $12$ and $3$ .

$- 243 a^{15} n^{8}$