Simplify ((2x^2u)/(z^-1))^-3(u^-4z^7)

${(\frac{2 x^{2} u}{z^{- 1}})}^{- 3} (u^{- 4} z^{7})$

Step 1

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

${(2 x^{2} u z)}^{- 3} (u^{- 4} z^{7})$

Step 2

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

$\frac{1}{{(2 x^{2} u z)}^{3}} (u^{- 4} z^{7})$

Step 3

Simplify the denominator.

Apply the product rule to $2 x^{2} u z$ .

$\frac{1}{{(2 x^{2} u)}^{3} z^{3}} (u^{- 4} z^{7})$

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

Apply the product rule to $2 x^{2} u$ .

$\frac{1}{{(2 x^{2})}^{3} u^{3} z^{3}} (u^{- 4} z^{7})$

Apply the product rule to $2 x^{2}$ .

$\frac{1}{2^{3} {(x^{2})}^{3} u^{3} z^{3}} (u^{- 4} z^{7})$

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

$\frac{1}{8 {(x^{2})}^{3} u^{3} z^{3}} (u^{- 4} z^{7})$

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

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

$\frac{1}{8 x^{2 \cdot 3} u^{3} z^{3}} (u^{- 4} z^{7})$

Multiply $2$ by $3$ .

$\frac{1}{8 x^{6} u^{3} z^{3}} (u^{- 4} z^{7})$

Step 4

Simplify terms.

Cancel the common factor of $u^{- 4} z^{3}$ .

Factor $u^{- 4} z^{3}$ out of $8 x^{6} u^{3} z^{3}$ .

$\frac{1}{u^{- 4} z^{3} (8 x^{6} u^{7})} (u^{- 4} z^{7})$

Factor $u^{- 4} z^{3}$ out of $u^{- 4} z^{7}$ .

$\frac{1}{u^{- 4} z^{3} (8 x^{6} u^{7})} (u^{- 4} z^{3} (z^{4}))$

Cancel the common factor.

$\frac{1}{u^{- 4} z^{3} (8 x^{6} u^{7})} (u^{- 4} z^{3} z^{4})$

Rewrite the expression.

$\frac{1}{8 x^{6} u^{7}} z^{4}$

Combine $\frac{1}{8 x^{6} u^{7}}$ and $z^{4}$ .

$\frac{z^{4}}{8 x^{6} u^{7}}$