Simplify (12q^-3)/((3p^4q^-3)^-4)

$\frac{12 q^{- 3}}{{(3 p^{4} q^{- 3})}^{- 4}}$

Simplify.

Move $q^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{{(3 p^{4} q^{- 3})}^{- 4} q^{3}}$

Move ${(3 p^{4} q^{- 3})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{12 {(3 p^{4} q^{- 3})}^{4}}{q^{3}}$

Simplify the numerator.

Apply the product rule to $3 p^{4} q^{- 3}$ .

$\frac{12 {(3 p^{4})}^{4} {(q^{- 3})}^{4}}{q^{3}}$

Apply the product rule to $3 p^{4}$ .

$\frac{12 (3^{4} {(p^{4})}^{4}) {(q^{- 3})}^{4}}{q^{3}}$

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

$\frac{12 (81 {(p^{4})}^{4}) {(q^{- 3})}^{4}}{q^{3}}$

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

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

$\frac{12 (81 p^{4 \cdot 4}) {(q^{- 3})}^{4}}{q^{3}}$

Multiply $4$ by $4$ .

$\frac{12 (81 p^{16}) {(q^{- 3})}^{4}}{q^{3}}$

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

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

$\frac{12 (81 p^{16}) q^{- 3 \cdot 4}}{q^{3}}$

Multiply $- 3$ by $4$ .

$\frac{12 (81 p^{16}) q^{- 12}}{q^{3}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12 \cdot 81 p^{16} \frac{1}{q^{12}}}{q^{3}}$

Combine exponents.

Multiply $12$ by $81$ .

$\frac{972 p^{16} \frac{1}{q^{12}}}{q^{3}}$

Combine $972$ and $\frac{1}{q^{12}}$ .

$\frac{p^{16} \frac{972}{q^{12}}}{q^{3}}$

Combine $p^{16}$ and $\frac{972}{q^{12}}$ .

$\frac{\frac{p^{16} \cdot 972}{q^{12}}}{q^{3}}$

Move $972$ to the left of $p^{16}$ .

$\frac{\frac{972 p^{16}}{q^{12}}}{q^{3}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{972 p^{16}}{q^{12}} \cdot \frac{1}{q^{3}}$

Combine.

$\frac{972 p^{16} \cdot 1}{q^{12} q^{3}}$

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

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

$\frac{972 p^{16} \cdot 1}{q^{12 + 3}}$

Add $12$ and $3$ .

$\frac{972 p^{16} \cdot 1}{q^{15}}$

Multiply $972 p^{16}$ by $1$ .

$\frac{972 p^{16}}{q^{15}}$