Simplify (-4x^3y^-1)^3*y^4

${(- 4 x^{3} y^{- 1})}^{3} \cdot y^{4}$

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

${(- 4 x^{3} \frac{1}{y})}^{3} \cdot y^{4}$

Multiply $- 4 x^{3} \frac{1}{y}$ .

Combine $- 4$ and $\frac{1}{y}$ .

${(x^{3} \frac{- 4}{y})}^{3} \cdot y^{4}$

Combine $x^{3}$ and $\frac{- 4}{y}$ .

${(\frac{x^{3} \cdot - 4}{y})}^{3} \cdot y^{4}$

Move $- 4$ to the left of $x^{3}$ .

${(\frac{- 4 \cdot x^{3}}{y})}^{3} \cdot y^{4}$

Move the negative in front of the fraction.

${(- \frac{(4) \cdot x^{3}}{y})}^{3} \cdot y^{4}$

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

Apply the product rule to $- \frac{4 x^{3}}{y}$ .

${(- 1)}^{3} {(\frac{4 x^{3}}{y})}^{3} \cdot y^{4}$

Apply the product rule to $\frac{4 x^{3}}{y}$ .

${(- 1)}^{3} \frac{{(4 x^{3})}^{3}}{y^{3}} \cdot y^{4}$

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

${(- 1)}^{3} \frac{4^{3} {(x^{3})}^{3}}{y^{3}} \cdot y^{4}$

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

$- \frac{4^{3} {(x^{3})}^{3}}{y^{3}} \cdot y^{4}$

Simplify the numerator.

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

$- \frac{64 {(x^{3})}^{3}}{y^{3}} \cdot y^{4}$

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

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

$- \frac{64 x^{3 \cdot 3}}{y^{3}} \cdot y^{4}$

Multiply $3$ by $3$ .

$- \frac{64 x^{9}}{y^{3}} \cdot y^{4}$

Cancel the common factor of $y^{3}$ .

Move the leading negative in $- \frac{64 x^{9}}{y^{3}}$ into the numerator.

$\frac{- 64 x^{9}}{y^{3}} \cdot y^{4}$

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

$\frac{- 64 x^{9}}{y^{3}} \cdot (y^{3} y)$

Cancel the common factor.

$\frac{- 64 x^{9}}{y^{3}} \cdot (y^{3} y)$

Rewrite the expression.

$- 64 x^{9} \cdot y$

$- 64 x^{9} y$