Simplify (-3u^-2v^-5)^-4

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

Step 1

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

${(- 3 \frac{1}{u^{2}} v^{- 5})}^{- 4}$

Step 2

Combine $- 3$ and $\frac{1}{u^{2}}$ .

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

Step 3

Move the negative in front of the fraction.

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

Step 4

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

${(- \frac{3}{u^{2}} \cdot \frac{1}{v^{5}})}^{- 4}$

Step 5

Multiply $\frac{1}{v^{5}}$ by $\frac{3}{u^{2}}$ .

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

Step 6

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

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

Step 7

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

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

${(- 1)}^{4} {(\frac{v^{5} u^{2}}{3})}^{4}$

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

${(- 1)}^{4} \frac{{(v^{5} u^{2})}^{4}}{3^{4}}$

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

${(- 1)}^{4} \frac{{(v^{5})}^{4} {(u^{2})}^{4}}{3^{4}}$

Step 8

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$1 \frac{{(v^{5})}^{4} {(u^{2})}^{4}}{3^{4}}$

Multiply $\frac{{(v^{5})}^{4} {(u^{2})}^{4}}{3^{4}}$ by $1$ .

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

Step 9

Simplify the numerator.

Multiply the exponents in ${(v^{5})}^{4}$ .

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

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

Multiply $5$ by $4$ .

$\frac{v^{20} {(u^{2})}^{4}}{3^{4}}$

Multiply the exponents in ${(u^{2})}^{4}$ .

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

$\frac{v^{20} u^{2 \cdot 4}}{3^{4}}$

Multiply $2$ by $4$ .

$\frac{v^{20} u^{8}}{3^{4}}$

Step 10

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

$\frac{v^{20} u^{8}}{81}$

Simplify (-3u^-2v^-5)^-4

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