Evaluate ((-6a^-9b^5)/(2a^2b^-4))^4

${(\frac{- 6 a^{- 9} b^{5}}{2 a^{2} b^{- 4}})}^{4}$

Move $a^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{- 6 b^{5}}{2 a^{2} b^{- 4} a^{9}})}^{4}$

Multiply $a^{2}$ by $a^{9}$ by adding the exponents.

Move $a^{9}$ .

${(\frac{- 6 b^{5}}{2 (a^{9} a^{2}) b^{- 4}})}^{4}$

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

${(\frac{- 6 b^{5}}{2 a^{9 + 2} b^{- 4}})}^{4}$

Add $9$ and $2$ .

${(\frac{- 6 b^{5}}{2 a^{11} b^{- 4}})}^{4}$

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{- 6 b^{5} b^{4}}{2 a^{11}})}^{4}$

Multiply $b^{5}$ by $b^{4}$ by adding the exponents.

Move $b^{4}$ .

${(\frac{- 6 (b^{4} b^{5})}{2 a^{11}})}^{4}$

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

${(\frac{- 6 b^{4 + 5}}{2 a^{11}})}^{4}$

Add $4$ and $5$ .

${(\frac{- 6 b^{9}}{2 a^{11}})}^{4}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6 b^{9}$ .

${(\frac{2 (- 3 b^{9})}{2 a^{11}})}^{4}$

Cancel the common factors.

Factor $2$ out of $2 a^{11}$ .

${(\frac{2 (- 3 b^{9})}{2 (a^{11})})}^{4}$

Cancel the common factor.

${(\frac{2 (- 3 b^{9})}{2 a^{11}})}^{4}$

Rewrite the expression.

${(\frac{- 3 b^{9}}{a^{11}})}^{4}$

Move the negative in front of the fraction.

${(- \frac{(3) b^{9}}{a^{11}})}^{4}$

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

Apply the product rule to $- \frac{3 b^{9}}{a^{11}}$ .

${(- 1)}^{4} {(\frac{3 b^{9}}{a^{11}})}^{4}$

Apply the product rule to $\frac{3 b^{9}}{a^{11}}$ .

${(- 1)}^{4} \frac{{(3 b^{9})}^{4}}{{(a^{11})}^{4}}$

Apply the product rule to $3 b^{9}$ .

${(- 1)}^{4} \frac{3^{4} {(b^{9})}^{4}}{{(a^{11})}^{4}}$

Simplify the expression.

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

$1 \frac{3^{4} {(b^{9})}^{4}}{{(a^{11})}^{4}}$

Multiply $\frac{3^{4} {(b^{9})}^{4}}{{(a^{11})}^{4}}$ by $1$ .

$\frac{3^{4} {(b^{9})}^{4}}{{(a^{11})}^{4}}$

Simplify the numerator.

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

$\frac{81 {(b^{9})}^{4}}{{(a^{11})}^{4}}$

Multiply the exponents in ${(b^{9})}^{4}$ .

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

$\frac{81 b^{9 \cdot 4}}{{(a^{11})}^{4}}$

Multiply $9$ by $4$ .

$\frac{81 b^{36}}{{(a^{11})}^{4}}$

Multiply the exponents in ${(a^{11})}^{4}$ .

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

$\frac{81 b^{36}}{a^{11 \cdot 4}}$

Multiply $11$ by $4$ .

$\frac{81 b^{36}}{a^{44}}$

Evaluate ((-6a^-9b^5)/(2a^2b^-4))^4

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