Simplify ((2u^4)/(4uv^-5))^-3

${(\frac{2 u^{4}}{4 u v^{- 5}})}^{- 3}$

Move $v^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 u^{4} v^{5}}{4 u})}^{- 3}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 u^{4} v^{5}$ .

${(\frac{2 (u^{4} v^{5})}{4 u})}^{- 3}$

Cancel the common factors.

Factor $2$ out of $4 u$ .

${(\frac{2 (u^{4} v^{5})}{2 (2 u)})}^{- 3}$

Cancel the common factor.

${(\frac{2 (u^{4} v^{5})}{2 (2 u)})}^{- 3}$

Rewrite the expression.

${(\frac{u^{4} v^{5}}{2 u})}^{- 3}$

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

Factor $u$ out of $u^{4} v^{5}$ .

${(\frac{u (u^{3} v^{5})}{2 u})}^{- 3}$

Cancel the common factors.

Factor $u$ out of $2 u$ .

${(\frac{u (u^{3} v^{5})}{u \cdot 2})}^{- 3}$

Cancel the common factor.

${(\frac{u (u^{3} v^{5})}{u \cdot 2})}^{- 3}$

Rewrite the expression.

${(\frac{u^{3} v^{5}}{2})}^{- 3}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{2}{u^{3} v^{5}})}^{3}$

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

Apply the product rule to $\frac{2}{u^{3} v^{5}}$ .

$\frac{2^{3}}{{(u^{3} v^{5})}^{3}}$

Apply the product rule to $u^{3} v^{5}$ .

$\frac{2^{3}}{{(u^{3})}^{3} {(v^{5})}^{3}}$

Raise $2$ to the power of $3$ .

$\frac{8}{{(u^{3})}^{3} {(v^{5})}^{3}}$

Simplify the denominator.

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

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

$\frac{8}{u^{3 \cdot 3} {(v^{5})}^{3}}$

Multiply $3$ by $3$ .

$\frac{8}{u^{9} {(v^{5})}^{3}}$

Multiply the exponents in ${(v^{5})}^{3}$ .

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

$\frac{8}{u^{9} v^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{8}{u^{9} v^{15}}$