Simplify ((m^-1n^4)^-1)/((m^2n)^-3)

$\frac{{(m^{- 1} n^{4})}^{- 1}}{{(m^{2} n)}^{- 3}}$

Simplify the expression.

Move ${(m^{- 1} n^{4})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(m^{2} n)}^{- 3} (m^{- 1} n^{4})}$

Move ${(m^{2} n)}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(m^{2} n)}^{3}}{m^{- 1} n^{4}}$

Move $m^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(m^{2} n)}^{3} m}{n^{4}}$

Simplify the numerator.

Apply the product rule to $m^{2} n$ .

$\frac{{(m^{2})}^{3} n^{3} m}{n^{4}}$

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

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

$\frac{m^{2 \cdot 3} n^{3} m}{n^{4}}$

Multiply $2$ by $3$ .

$\frac{m^{6} n^{3} m}{n^{4}}$

Combine exponents.

Raise $m$ to the power of $1$ .

$\frac{m^{1} m^{6} n^{3}}{n^{4}}$

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

$\frac{m^{1 + 6} n^{3}}{n^{4}}$

Add $1$ and $6$ .

$\frac{m^{7} n^{3}}{n^{4}}$

Cancel the common factor of $n^{3}$ and $n^{4}$ .

Factor $n^{3}$ out of $m^{7} n^{3}$ .

$\frac{n^{3} m^{7}}{n^{4}}$

Cancel the common factors.

Factor $n^{3}$ out of $n^{4}$ .

$\frac{n^{3} m^{7}}{n^{3} n}$

Cancel the common factor.

$\frac{n^{3} m^{7}}{n^{3} n}$

Rewrite the expression.

$\frac{m^{7}}{n}$