Simplify ((-3a^-2b^-3)/(a^-5b))^-4

${(\frac{- 3 a^{- 2} b^{- 3}}{a^{- 5} b})}^{- 4}$

Step 1

Move $b^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{- 3 a^{- 2}}{a^{- 5} b \cdot b^{3}})}^{- 4}$

Step 2

Multiply $b$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

${(\frac{- 3 a^{- 2}}{a^{- 5} (b^{3} b)})}^{- 4}$

Multiply $b^{3}$ by $b$ .

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

${(\frac{- 3 a^{- 2}}{a^{- 5} (b^{3} b^{1})})}^{- 4}$

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

${(\frac{- 3 a^{- 2}}{a^{- 5} b^{3 + 1}})}^{- 4}$

Add $3$ and $1$ .

${(\frac{- 3 a^{- 2}}{a^{- 5} b^{4}})}^{- 4}$

Step 3

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

${(\frac{- 3 a^{- 2} a^{5}}{b^{4}})}^{- 4}$

Step 4

Multiply $a^{- 2}$ by $a^{5}$ by adding the exponents.

Move $a^{5}$ .

${(\frac{- 3 (a^{5} a^{- 2})}{b^{4}})}^{- 4}$

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

${(\frac{- 3 a^{5 - 2}}{b^{4}})}^{- 4}$

Subtract $2$ from $5$ .

${(\frac{- 3 a^{3}}{b^{4}})}^{- 4}$

Step 5

Move the negative in front of the fraction.

${(- \frac{(3) a^{3}}{b^{4}})}^{- 4}$

Step 6

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

${(- \frac{b^{4}}{3 a^{3}})}^{4}$

Step 7

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

Apply the product rule to $- \frac{b^{4}}{3 a^{3}}$ .

${(- 1)}^{4} {(\frac{b^{4}}{3 a^{3}})}^{4}$

Apply the product rule to $\frac{b^{4}}{3 a^{3}}$ .

${(- 1)}^{4} \frac{{(b^{4})}^{4}}{{(3 a^{3})}^{4}}$

Apply the product rule to $3 a^{3}$ .

${(- 1)}^{4} \frac{{(b^{4})}^{4}}{3^{4} {(a^{3})}^{4}}$

Step 8

Simplify the expression.

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

$1 \frac{{(b^{4})}^{4}}{3^{4} {(a^{3})}^{4}}$

Multiply $\frac{{(b^{4})}^{4}}{3^{4} {(a^{3})}^{4}}$ by $1$ .

$\frac{{(b^{4})}^{4}}{3^{4} {(a^{3})}^{4}}$

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

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

$\frac{b^{4 \cdot 4}}{3^{4} {(a^{3})}^{4}}$

Multiply $4$ by $4$ .

$\frac{b^{16}}{3^{4} {(a^{3})}^{4}}$

Step 9

Simplify the denominator.

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

$\frac{b^{16}}{81 {(a^{3})}^{4}}$

Multiply the exponents in ${(a^{3})}^{4}$ .

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

$\frac{b^{16}}{81 a^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{b^{16}}{81 a^{12}}$

Simplify ((-3a^-2b^-3)/(a^-5b))^-4

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